Chapter 19 : Indefinite Integrals - Rd Sharma Solutions for Class 12-science Maths CBSE

Chapter 19 - Indefinite Integrals Excercise Ex. 19.1

Question 1

integrate each of the following functions with respect to x:

begin mathsize 11px style open parentheses i close parentheses integral x to the power of 4 d x
open parentheses i i close parentheses integral space x to the power of 5 divided by 4 end exponent d x
open parentheses i i i close parentheses integral 1 over x to the power of 5 d x
open parentheses i v close parentheses integral 1 over x to the power of 3 divided by 2 end exponent d x
open parentheses v close parentheses integral 3 to the power of x d x
open parentheses v i close parentheses integral fraction numerator 1 over denominator cube root of x squared end root end fraction space d x
open parentheses v i i close parentheses integral 3 to the power of 2 log 3 x end exponent space d x
open parentheses v i i i close parentheses integral log subscript x x space d x end style

Solution 1



Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6

Chapter 19 - Indefinite Integrals Excercise Ex. 19.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20

Solution 20

Question 21

Solution 21

Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25

Solution 25

Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solution 39
Question 40
Solution 40
Question 41
Solution 41
Question 42

Evaluate :integral fraction numerator cos x over denominator 1 plus cos x end fraction d x

Solution 42

Question 43

Evalute :

integral fraction numerator 1 minus cos x over denominator 1 plus cos x end fraction d x

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Chapter 19 - Indefinite Integrals Excercise Ex. 19.3

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Evaluate the following integrals:

integral fraction numerator 1 over denominator cos squared x open parentheses 1 plus tan x close parentheses squared end fraction

Solution 19

Chapter 19 - Indefinite Integrals Excercise Ex. 19.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Chapter 19 - Indefinite Integrals Excercise Ex. 19.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 19 - Indefinite Integrals Excercise Ex. 19.6

Question 1

Solution 1

Question 2

Evalute the following integral :integral sin cubed left parenthesis 2 x plus 1 right parenthesis d x

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 19 - Indefinite Integrals Excercise Ex. 19.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 19 - Indefinite Integrals Excercise Ex. 19.8

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Chapter 19 - Indefinite Integrals Excercise Ex. 19.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20


Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

L e t space tan to the power of minus 1 end exponent x equals t
D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o comma space w comma space w e space h a v e comma
fraction numerator 1 over denominator 1 plus x squared end fraction d x equals d t
rightwards double arrow integral fraction numerator e to the power of m tan to the power of minus 1 end exponent x end exponent over denominator 1 plus x squared end fraction equals integral e to the power of m t end exponent cross times d t
rightwards double arrow integral fraction numerator e to the power of m tan to the power of minus 1 end exponent x end exponent over denominator 1 plus x squared end fraction equals e to the power of m t end exponent over m
R e s u b s t i t u t i n g space t h e space v a l u e space o f space t space i n space t h e space a b o v e space s o l u t i o n comma space w e space h a v e comma
space space space space space space rightwards double arrow integral fraction numerator e to the power of m tan to the power of minus 1 end exponent x end exponent over denominator 1 plus x squared end fraction equals e to the power of m tan to the power of minus 1 end exponent x end exponent over m plus C

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63



Question 64

Solution 64

Question 65

Solution 65

Question 66

Solution 66

Question 67

Solution 67

Question 68

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Chapter 19 - Indefinite Integrals Excercise Ex. 19.10

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4


Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 19 - Indefinite Integrals Excercise Ex. 19.11

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 19 - Indefinite Integrals Excercise Ex. 19.12

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Chapter 19 - Indefinite Integrals Excercise Ex. 19.14

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 19 - Indefinite Integrals Excercise Ex. 19.15

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 19 - Indefinite Integrals Excercise Ex. 19.16

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 19 - Indefinite Integrals Excercise Ex. 19.17

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 19 - Indefinite Integrals Excercise Ex. 19.18

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Chapter 19 - Indefinite Integrals Excercise Ex. 19.19

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

 

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Evaluate the following integrals:

Solution 10

Question 11

Solution 11

Question 12

Evaluate the following integrals:

Solution 12

Chapter 19 - Indefinite Integrals Excercise Ex. 19.20

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

 

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 19 - Indefinite Integrals Excercise Ex. 19.21

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Evaluate the following integral

integral fraction numerator 5 x plus 3 over denominator square root of x squared plus 4 x plus 10 end root end fraction d x

Solution 17

Question 18

Evaluate the following integral:

Solution 18

Chapter 19 - Indefinite Integrals Excercise Ex. 19.22

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 19 - Indefinite Integrals Excercise Ex. 19.23

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Chapter 19 - Indefinite Integrals Excercise Ex. 19.24

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3


Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 19 - Indefinite Integrals Excercise Ex. 19.25

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Evaluate the following integral:

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

begin mathsize 12px style Evaluate space the space integrals colon
integral sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses dx end style

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38


Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

            

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

begin mathsize 12px style Evaluate space the space integrals colon
integral tan to the power of negative 1 end exponent square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root dx end style

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Chapter 19 - Indefinite Integrals Excercise Ex. 19.26

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

begin mathsize 12px style Evaluagte space the space following space integrals colon
integral straight e to the power of straight x open parentheses fraction numerator straight x minus 1 over denominator 2 straight x squared end fraction close parentheses dx end style

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Evaluate the following integral:

Solution 24

Chapter 19 - Indefinite Integrals Excercise Ex. 19.27

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 19 - Indefinite Integrals Excercise Ex. 19.28

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Chapter 19 - Indefinite Integrals Excercise Ex. 19.29

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Evaluate the following integral:

Solution 11

Question 12

Evaluate the following integral:

Solution 12

Chapter 19 - Indefinite Integrals Excercise Ex. 19.30

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Evaluate the following integral:

 

Solution 15

  

Question 16

Solution 16

Question 17

Evaluate the following integral:

 

  

Solution 17

  

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Evaluate the following integral:

 

  

Solution 24

  

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Evaluate the following integral:

 

Solution 46

  

Question 47

Solution 47

Question 48

Solution 48

Question 49

Evaluate the following integral:

 

  

Solution 49

  

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59


Question 60

Solution 60

Question 61

Solution 61

Chapter 19 - Indefinite Integrals Excercise Ex. 19.31

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Evaluate the following integral:

Solution 11

Chapter 19 - Indefinite Integrals Excercise Ex. 19.32

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 19 - Indefinite Integrals Excercise MCQ

Question 1

Mark the correct alternative in each of the following:

begin mathsize 12px style integral fraction numerator straight x over denominator 4 plus straight x to the power of 4 end fraction dx space is space equal space to

open parentheses straight a close parentheses space 1 fourth tan to the power of negative 1 end exponent straight x squared space plus space straight C
open parentheses straight b close parentheses space 1 fourth tan to the power of negative 1 end exponent open parentheses straight x squared over 2 close parentheses
open parentheses straight c close parentheses space 1 half tan to the power of negative 1 end exponent open parentheses straight x squared over 2 close parentheses
open parentheses straight d close parentheses space none space of space these end style

Solution 1

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
straight I equals integral fraction numerator straight x over denominator 4 plus straight x to the power of 4 end fraction dx space
Put space straight x squared equals straight t
rightwards double arrow 2 xdx equals dt
rightwards double arrow xdx equals dt over 2
straight I equals integral fraction numerator begin display style dt over 2 end style over denominator 4 plus straight t squared end fraction
straight I equals 1 half tan to the power of negative 1 end exponent open parentheses straight t over 2 close parentheses plus straight C
straight I equals 1 half tan to the power of negative 1 end exponent open parentheses straight x squared over 2 close parentheses plus straight C
end style

Question 2

Mark the correct alternative in each of the following:

begin mathsize 12px style integral fraction numerator 1 over denominator cos space straight x space plus square root of 3 space sin space straight x end fraction dx space is space equal space to

open parentheses straight a close parentheses space log space tan open parentheses straight pi over 3 plus straight x over 2 close parentheses plus straight C
open parentheses straight b close parentheses space log space tan open parentheses straight x over 2 minus straight pi over 3 close parentheses plus straight C
open parentheses straight c close parentheses space 1 half space log space tan open parentheses straight x over 2 plus straight pi over 3 close parentheses plus straight C
open parentheses straight d close parentheses space none space of space these end style

Solution 2

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
straight I equals integral fraction numerator 1 over denominator cosx plus square root of 3 sinx end fraction dx
straight I equals 1 half integral fraction numerator 1 over denominator begin display style cosx over 2 end style plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style sinx end fraction dx
straight I equals 1 half integral fraction numerator 1 over denominator cos open parentheses straight x minus begin display style straight pi over 6 end style close parentheses end fraction dx
straight I equals 1 half integral sec open parentheses straight x minus straight pi over 6 close parentheses dx
straight I equals 1 half ln open vertical bar tan open parentheses straight x over 2 plus straight pi over 3 close parentheses close vertical bar plus straight C end style

Question 3

Mark the correct alternative in each of the following:

begin mathsize 12px style integral straight x space sec space straight x squared space dx space is space equal space to

open parentheses straight a close parentheses space 1 half log open parentheses sec space straight x squared space plus space tan space straight x squared close parentheses plus straight C
open parentheses straight b close parentheses space straight x squared over 2 log open parentheses sec space straight x squared space plus space tan space straight x squared close parentheses plus straight C
open parentheses straight c close parentheses space 2 space log open parentheses sec space straight x squared space plus space tan space straight x squared close parentheses plus straight C

open parentheses straight d close parentheses space none space of space these
end style

Solution 3

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
straight I equals integral xsecx squared dx
Put space straight x squared equals straight t rightwards double arrow straight x equals square root of straight t
rightwards double arrow 2 xdx equals dt
rightwards double arrow xdx equals dt over 2
straight I equals integral sect dt over 2
straight I equals 1 half log open vertical bar sect plus tant close vertical bar plus straight C
straight I equals 1 half log open vertical bar sec straight x squared plus tan straight x squared close vertical bar plus straight C
end style

Question 4

begin mathsize 12px style If space integral fraction numerator 1 over denominator 5 plus 4 space sin space straight x end fraction dx equals straight A space tan to the power of negative 1 end exponent open parentheses straight B space tan straight x over 2 plus 4 over 3 close parentheses plus straight C comma space then

open parentheses straight a close parentheses space straight A equals 2 over 3 comma space straight B equals 5 over 3
open parentheses straight b close parentheses space straight A equals 1 third comma space straight B equals 2 over 3
open parentheses straight c close parentheses space straight A equals negative 2 over 3 comma space straight B equals 5 over 3
open parentheses straight d close parentheses space straight A equals 1 third comma space straight B equals negative 5 over 3 end style

Solution 4

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
integral fraction numerator 1 over denominator 5 plus 4 sinx end fraction dx equals Atan to the power of negative 1 end exponent open parentheses Btan straight x over 2 plus 4 over 3 close parentheses plus straight C
Put space tan straight x over 2 equals straight t rightwards double arrow straight x equals 2 tan to the power of negative 1 end exponent straight t
rightwards double arrow dx equals fraction numerator 2 dt over denominator 1 plus straight t squared end fraction
rightwards double arrow sinx equals fraction numerator 2 tan begin display style straight x over 2 end style over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction
integral fraction numerator 1 over denominator 5 plus 4 sinx end fraction dx
equals integral fraction numerator begin display style fraction numerator 2 dt over denominator 1 plus straight t squared end fraction end style over denominator 5 plus 4 cross times fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end fraction
equals integral fraction numerator 2 dt over denominator 5 straight t squared plus 8 straight t plus 5 end fraction
equals 2 over 5 integral fraction numerator dt over denominator straight t squared plus begin display style 8 over 5 end style straight t plus 1 end fraction
Using space completing space square space method
we space get
straight I equals 2 over 3 tan to the power of negative 1 end exponent open parentheses 5 over 3 tan straight x over 2 plus 4 over 3 close parentheses plus straight C
straight A equals 2 over 3 space and space straight B equals 5 over 3
end style

Question 5

begin mathsize 12px style integral straight x to the power of sin space straight x end exponent open parentheses fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x close parentheses dx space is space equal space to

open parentheses straight a close parentheses space straight x to the power of sin space straight x end exponent plus straight C

open parentheses straight b close parentheses space straight x to the power of sin space straight x end exponent space cos space straight x plus straight C
open parentheses straight c close parentheses space open parentheses straight x to the power of sin space straight x end exponent close parentheses squared over 2 plus straight C
open parentheses straight d close parentheses space none space of space these end style

Solution 5

begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis
straight I equals integral straight x to the power of sinx open parentheses sinx over straight x plus cosxlogx close parentheses dx
Put space straight x to the power of sinx equals straight t
taking space log space on space both space sides comma
logt equals sinxlogx
1 over straight t dt equals sinx over straight x plus cosxlogx
rightwards double arrow straight I equals integral straight t cross times dt over straight t
straight I equals straight t plus straight C
straight I equals straight x to the power of sinx plus straight C
end style

Question 6

begin mathsize 12px style Integration space of space fraction numerator 1 over denominator 1 plus open parentheses log subscript straight e straight x close parentheses squared end fraction with space respect space to space log subscript straight e straight x space is

open parentheses straight a close parentheses space fraction numerator tan to the power of negative 1 end exponent open parentheses log subscript straight e straight x close parentheses over denominator straight x end fraction plus straight C
open parentheses straight b close parentheses space tan to the power of negative 1 end exponent open parentheses log subscript straight e straight x close parentheses space plus space straight C
open parentheses straight c close parentheses space fraction numerator tan to the power of negative 1 end exponent straight x over denominator straight x end fraction plus straight C
open parentheses straight d close parentheses space none space of space these end style

Solution 6

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
integral fraction numerator 1 over denominator 1 plus open parentheses log subscript straight e straight x close parentheses squared end fraction straight d open parentheses log subscript straight e straight x close parentheses
Put space log subscript straight e straight x equals straight t
integral fraction numerator dt over denominator 1 plus straight t squared end fraction equals tan to the power of negative 1 end exponent straight t plus straight C equals tan to the power of negative 1 end exponent open parentheses space log subscript straight e straight x close parentheses plus straight C
end style

Question 7

begin mathsize 12px style If space integral fraction numerator cos space 8 straight x plus 1 over denominator tan space 2 straight x space minus space cot space 2 straight x end fraction dx equals straight a space cos space 8 straight x space plus space straight C comma space then space straight a equals

open parentheses straight a close parentheses space minus 1 over 16
open parentheses straight b close parentheses space 1 over 8
open parentheses straight c close parentheses space 1 over 16
open parentheses straight d close parentheses space minus 1 over 8 end style

Solution 7

begin mathsize 12px style Correct space option colon thin space left parenthesis straight c right parenthesis
integral fraction numerator cos 8 straight x plus 1 over denominator tan 2 straight x minus cot 2 straight x end fraction dx
equals integral fraction numerator 2 cos squared 4 straight x over denominator begin display style fraction numerator sin 2 straight x over denominator cos 2 straight x end fraction end style minus begin display style fraction numerator cos 2 straight x over denominator sin 2 straight x end fraction end style end fraction dx
equals integral fraction numerator 2 cos squared 4 straight x over denominator sin squared 2 straight x minus cos squared 2 straight x end fraction cross times sin 2 xcos 2 xdx
equals integral negative fraction numerator cos squared 4 xsin 4 straight x over denominator cos 4 straight x end fraction dx
equals fraction numerator negative 1 over denominator 2 end fraction integral sin 8 xdx
equals fraction numerator cos 8 straight x over denominator 16 end fraction plus straight C
straight a equals 1 over 16 end style

Question 8

begin mathsize 12px style If space integral fraction numerator sin to the power of 8 space straight x space minus space cos to the power of 8 space straight x over denominator 1 minus 2 space sin squared space straight x space cos squared space straight x end fraction dx equals straight a space sin space 2 straight x space plus space straight C comma space then space straight a equals

open parentheses straight a close parentheses space minus 1 divided by 2

open parentheses straight b close parentheses space 1 divided by 2

open parentheses straight c close parentheses space minus 1

open parentheses straight d close parentheses space 1 end style

Solution 8

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
integral fraction numerator sin to the power of 8 straight x minus cos to the power of 8 straight x over denominator 1 minus 2 sin squared xcos squared straight x end fraction dx
integral fraction numerator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses open parentheses sin to the power of 4 straight x minus cos to the power of 4 straight x close parentheses over denominator 1 minus 2 sin squared xcos squared straight x end fraction dx
integral fraction numerator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses open parentheses sin squared straight x plus cos squared straight x close parentheses open parentheses sin squared straight x minus cos squared straight x close parentheses over denominator open parentheses sin squared straight x plus cos squared straight x close parentheses squared minus 2 sin squared xcos squared straight x end fraction dx
integral fraction numerator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses open parentheses sin squared straight x minus cos squared straight x close parentheses over denominator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses end fraction dx
integral negative cos 2 xdx
fraction numerator negative sin 2 straight x over denominator 2 end fraction plus straight C
rightwards double arrow straight a equals fraction numerator negative 1 over denominator 2 end fraction end style

Question 9

begin mathsize 11px style integral open parentheses straight x minus 1 close parentheses straight e to the power of negative straight x end exponent space dx space is space equal space to

open parentheses straight a close parentheses space minus xe to the power of straight x plus straight C
open parentheses straight b close parentheses space xe to the power of straight x space plus space straight C
open parentheses straight c close parentheses space minus xe to the power of negative straight x end exponent plus straight C
open parentheses straight d close parentheses space xe to the power of negative straight x end exponent plus straight C
end style

Solution 9

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
integral open parentheses straight x minus 1 close parentheses straight e to the power of negative straight x end exponent dx
equals open parentheses straight x minus 1 close parentheses integral straight e to the power of negative straight x end exponent dx minus integral open parentheses open square brackets fraction numerator straight d open parentheses straight x minus 1 close parentheses over denominator dx end fraction close square brackets integral straight e to the power of negative straight x end exponent dx close parentheses dx
equals open parentheses straight x minus 1 close parentheses fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction minus integral fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction dx
equals negative open parentheses straight x minus 1 close parentheses straight e to the power of negative straight x end exponent plus fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction plus straight C
equals negative xe to the power of negative straight x end exponent plus straight e to the power of negative straight x end exponent minus straight e to the power of negative straight x end exponent plus straight C
equals negative xe to the power of negative straight x end exponent plus straight C
end style

Question 10

begin mathsize 11px style If space integral 2 to the power of 1 divided by straight x end exponent over straight x squared dx space equals space straight k space 2 to the power of 1 divided by straight x end exponent plus straight C comma space then space straight k space is space equal space to
open parentheses straight a close parentheses space minus fraction numerator 1 over denominator log subscript straight e 2 end fraction
open parentheses straight b close parentheses space minus log subscript straight e 2
open parentheses straight c close parentheses space minus 1
open parentheses straight d close parentheses space 1 half end style

Solution 10

begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis
straight I equals integral 2 to the power of begin display style 1 over straight x end style end exponent over straight x squared dx
Put space 1 over straight x equals straight t
rightwards double arrow fraction numerator negative 1 over denominator straight x squared end fraction dx equals dt rightwards double arrow 1 over straight x squared dx equals negative dt
straight I equals integral 2 to the power of straight t open parentheses negative dt close parentheses
straight I equals fraction numerator negative 2 to the power of straight t over denominator log subscript straight e 2 end fraction plus straight C
rightwards double arrow straight I equals fraction numerator negative 2 to the power of begin display style 1 over straight x end style end exponent over denominator log subscript straight e 2 end fraction plus straight C
straight k equals fraction numerator negative 1 over denominator log subscript straight e 2 end fraction
end style

Question 11

begin mathsize 11px style integral fraction numerator 1 over denominator 1 plus tan space straight x end fraction dx equals

open parentheses straight a close parentheses space log subscript straight e open parentheses straight x plus sin space straight x close parentheses plus straight C
open parentheses straight b close parentheses space log subscript straight e open parentheses sin space straight x plus cos space straight x close parentheses plus straight C
open parentheses straight c close parentheses space 2 space sec squared straight x over 2 plus straight C
open parentheses straight d close parentheses space 1 half open curly brackets straight x space plus space log open parentheses sin space straight x space plus space cos space straight x close parentheses close curly brackets space plus space straight C end style

Solution 11

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
straight I equals integral fraction numerator 1 over denominator 1 plus tanx end fraction dx
equals integral fraction numerator cosx over denominator sinx plus cosx end fraction dx
Numerator space can space be space written space as colon
cosx equals straight A open parentheses sinx plus cosx close parentheses plus straight B fraction numerator straight d open parentheses sinx plus cosx close parentheses over denominator dx end fraction
cosx equals open parentheses straight A minus straight B close parentheses sinx plus open parentheses straight A plus straight B close parentheses cosx
rightwards double arrow straight A minus straight B equals 0 space and space straight A plus straight B equals 1
rightwards double arrow straight A equals 1 half equals straight B
straight I equals integral fraction numerator open square brackets begin display style 1 half end style open parentheses sinx plus cosx close parentheses plus 1 half open parentheses cosx minus sinx close parentheses close square brackets dx over denominator sinx plus cosx end fraction
straight I equals 1 half integral open parentheses 1 plus fraction numerator cosx minus sinx over denominator sinx plus cosx end fraction close parentheses dx
straight I equals 1 half open square brackets 1 plus ln open parentheses sinx plus cosx close parentheses close square brackets plus straight C end style

Question 12

begin mathsize 12px style integral open vertical bar straight x close vertical bar cubed space dx space is space equal space to

open parentheses straight a close parentheses space fraction numerator negative straight x to the power of 4 over denominator 4 end fraction plus straight C
open parentheses straight b close parentheses space open vertical bar straight x close vertical bar to the power of 4 over 4 plus straight C
open parentheses straight c close parentheses space straight x to the power of 4 over 4 plus straight C
open parentheses straight d close parentheses space none space of space these end style

Solution 12

begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis
integral open vertical bar straight x close vertical bar cubed dx
If space straight x greater than 0
rightwards double arrow integral straight x cubed dx
equals straight x to the power of 4 over 4 plus straight C
If space straight x less than 0
rightwards double arrow integral negative straight x cubed dx
equals negative straight x to the power of 4 over 4 plus straight C end style

Question 13

begin mathsize 12px style The space value space of space integral fraction numerator cos space square root of straight x over denominator square root of straight x end fraction dx space is
open parentheses straight a close parentheses space 2 space cos square root of straight x plus straight C
open parentheses straight b close parentheses space square root of fraction numerator cos space straight x over denominator straight x end fraction end root plus straight C
open parentheses straight c close parentheses space sin space square root of straight x plus straight C
open parentheses straight d close parentheses space 2 space sin space square root of straight x plus space straight C end style

Solution 13

begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis
straight I equals integral fraction numerator cos square root of straight x over denominator square root of straight x end fraction dx
Put comma space
square root of straight x space equals straight t
fraction numerator 1 over denominator 2 square root of straight x end fraction dx equals dt
rightwards double arrow fraction numerator 1 over denominator square root of straight x end fraction dx equals 2 dt
straight I equals integral cost space 2 dt
straight I equals 2 sint plus straight C equals 2 sin square root of straight x plus straight C end style

Question 14

begin mathsize 12px style integral straight e to the power of straight x open parentheses 1 minus cot space straight x space plus space cot squared straight x close parentheses space dx equals

open parentheses straight a close parentheses space straight e to the power of straight x space cot space straight x space plus space straight C
open parentheses straight b close parentheses space minus straight e to the power of straight x space cot space straight x space plus space straight C
open parentheses straight c close parentheses space straight e to the power of straight x space cosec space straight x space plus space straight C
open parentheses straight d close parentheses space minus straight e to the power of straight x space cosec space straight x space plus space straight C end style

Solution 14

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
straight I equals integral straight e to the power of straight x open parentheses 1 minus cotx plus cot squared straight x close parentheses dx
straight I equals integral straight e to the power of straight x open parentheses 1 plus cot squared straight x minus cotx close parentheses dx
straight I equals integral straight e to the power of straight x open parentheses cosec squared straight x minus cotx close parentheses dx
Here comma space straight f left parenthesis straight x right parenthesis equals negative cotx rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals cosec squared straight x
straight I equals negative straight e to the power of straight x cotx plus straight C end style

Question 15

Error converting from MathML to accessible text.

Solution 15

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
straight I equals integral fraction numerator sin to the power of 6 xdx over denominator cos to the power of 8 straight x end fraction
straight I equals integral tan to the power of 6 xsec squared xdx
Put space tanx equals straight t rightwards double arrow sec squared xdx equals dt
straight I equals integral straight t to the power of 6 dt
straight I equals straight t to the power of 7 over 7 plus straight C
straight I equals fraction numerator tan to the power of 7 straight x over denominator 7 end fraction plus straight C end style

Question 16

begin mathsize 12px style integral fraction numerator 1 over denominator 7 plus 5 cos space straight x end fraction dx equals

open parentheses straight a close parentheses space fraction numerator 1 over denominator square root of 6 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 6 end fraction tan straight x over 2 close parentheses plus straight C
open parentheses straight b close parentheses space fraction numerator begin display style 1 end style over denominator begin display style square root of 3 end style end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator begin display style 1 end style over denominator begin display style square root of 3 end style end fraction tan fraction numerator begin display style straight x end style over denominator begin display style 2 end style end fraction close parentheses plus straight C
open parentheses straight c close parentheses space fraction numerator begin display style 1 end style over denominator begin display style 4 end style end fraction tan to the power of negative 1 end exponent open parentheses tan fraction numerator begin display style straight x end style over denominator begin display style 2 end style end fraction close parentheses plus straight C
open parentheses straight d close parentheses space fraction numerator begin display style 1 end style over denominator begin display style 7 end style end fraction tan to the power of negative 1 end exponent open parentheses tan fraction numerator begin display style straight x end style over denominator begin display style 2 end style end fraction close parentheses plus straight C end style

Solution 16

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
straight I equals integral fraction numerator dx over denominator 7 plus 5 cosx end fraction
put space tan straight x over 2 equals straight t rightwards double arrow dx equals fraction numerator 2 dt over denominator 1 plus straight t squared end fraction
rightwards double arrow cosx equals fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction
straight I equals integral fraction numerator fraction numerator 2 dt over denominator 1 plus straight t squared end fraction over denominator 7 plus 5 cross times fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end fraction
rightwards double arrow straight I equals integral fraction numerator 1 over denominator straight t squared plus 6 end fraction dt
straight I equals fraction numerator 1 over denominator square root of 6 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight t over denominator square root of 6 end fraction close parentheses plus straight C
rightwards double arrow fraction numerator 1 over denominator square root of 6 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator tan begin display style straight x over 2 end style over denominator square root of 6 end fraction close parentheses plus straight C
end style

Question 17

begin mathsize 12px style integral fraction numerator 1 over denominator 1 minus cosx minus sinx end fraction dx equals
open parentheses straight a close parentheses space log space open vertical bar 1 plus cot straight x over 2 close vertical bar plus straight C
open parentheses straight b close parentheses space log space open vertical bar 1 minus tan fraction numerator begin display style straight x end style over denominator begin display style 2 end style end fraction close vertical bar plus straight C
open parentheses straight c close parentheses space log space open vertical bar 1 minus cot fraction numerator begin display style straight x end style over denominator begin display style 2 end style end fraction close vertical bar plus straight C
open parentheses straight d close parentheses space log space open vertical bar 1 plus tan fraction numerator begin display style straight x end style over denominator begin display style 2 end style end fraction close vertical bar plus straight C end style

Solution 17

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
integral fraction numerator dx over denominator 1 minus cosx minus sinx end fraction
Put space straight t equals space tan straight x over 2 rightwards double arrow dx equals fraction numerator 2 dt over denominator 1 plus straight t squared end fraction
cosx equals fraction numerator 1 minus tan squared straight x over 2 over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction
sinx equals fraction numerator 2 tan straight x over 2 over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction
put space in space the space question
straight I equals integral fraction numerator begin display style fraction numerator 2 dt over denominator 1 plus straight t squared end fraction end style over denominator 1 minus fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction minus fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end fraction
straight I equals integral fraction numerator dt over denominator straight t squared minus straight t end fraction
straight I equals integral fraction numerator dt over denominator straight t squared minus straight t plus begin display style 1 fourth end style minus begin display style 1 fourth end style end fraction
rightwards double arrow straight I equals ln open vertical bar 1 minus cot straight x over 2 close vertical bar plus straight C

end style

Question 18

begin mathsize 12px style integral fraction numerator straight x plus 3 over denominator open parentheses straight x plus 4 close parentheses squared end fraction straight e to the power of straight x dx equals
open parentheses straight a close parentheses space fraction numerator straight e to the power of straight x over denominator straight x plus 4 end fraction plus straight C
open parentheses straight b close parentheses space fraction numerator begin display style straight e to the power of straight x end style over denominator begin display style straight x plus 3 end style end fraction plus straight C
open parentheses straight c close parentheses space fraction numerator begin display style 1 end style over denominator begin display style open parentheses straight x plus 4 close parentheses squared end style end fraction plus straight C
open parentheses straight d close parentheses space fraction numerator begin display style straight e to the power of straight x end style over denominator begin display style open parentheses straight x plus 4 close parentheses squared end style end fraction plus straight C end style

Solution 18

begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis
straight I equals integral fraction numerator straight x plus 3 over denominator open parentheses straight x plus 4 close parentheses squared end fraction straight e to the power of straight x dx
straight I equals integral open parentheses fraction numerator straight x plus 4 minus 1 over denominator open parentheses straight x plus 4 close parentheses squared end fraction close parentheses straight e to the power of straight x dx
straight I equals integral open parentheses fraction numerator 1 over denominator straight x plus 4 end fraction minus 1 over open parentheses straight x plus 4 close parentheses squared close parentheses straight e to the power of straight x dx
straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator straight x plus 4 end fraction rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals negative 1 over open parentheses straight x plus 4 close parentheses squared
rightwards double arrow straight I equals fraction numerator straight e to the power of straight x over denominator straight x plus 4 end fraction plus straight C
end style

Question 19

begin mathsize 12px style integral fraction numerator sin space straight x over denominator 3 space plus space 4 space cos squared straight x end fraction dx
open parentheses straight a close parentheses space log space open parentheses 3 plus 4 cos squared straight x close parentheses plus straight C
open parentheses straight b close parentheses space fraction numerator 1 over denominator 2 square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator cos begin display style space end style begin display style straight x end style over denominator square root of 3 end fraction close parentheses plus straight C
open parentheses straight c close parentheses space minus fraction numerator begin display style 1 end style over denominator begin display style 2 square root of 3 end style end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator begin display style 2 space cos space straight x end style over denominator square root of 3 end fraction close parentheses plus straight C
open parentheses straight d close parentheses space fraction numerator begin display style 1 end style over denominator begin display style 2 square root of 3 end style end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator begin display style 2 space cos space straight x end style over denominator square root of 3 end fraction close parentheses plus straight C end style

Solution 19

begin mathsize 12px style Correct space option colon thin space left parenthesis straight c right parenthesis
straight I equals integral fraction numerator sinx over denominator 3 plus 4 cos squared straight x end fraction dx
Put space cosx equals straight t
rightwards double arrow negative sinxdx equals dt
rightwards double arrow sinxdx equals negative dt
straight I equals integral fraction numerator negative dt over denominator 3 plus 4 straight t squared end fraction
straight I equals fraction numerator negative 1 over denominator 2 square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight t over denominator square root of 3 end fraction close parentheses plus straight C
straight I equals fraction numerator negative 1 over denominator 2 square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator 2 cosx over denominator square root of 3 end fraction close parentheses plus straight C end style

Question 20

begin mathsize 12px style integral straight e to the power of straight x open parentheses fraction numerator 1 minus sin space straight x over denominator 1 minus cos space straight x end fraction close parentheses dx
open parentheses straight a close parentheses space minus straight e to the power of straight x space tan space straight x over 2 plus straight C
open parentheses straight b close parentheses space minus straight e to the power of straight x space cot space straight x over 2 plus straight C
open parentheses straight c close parentheses space minus 1 half space straight e to the power of straight x space tan space straight x over 2 plus straight C
open parentheses straight a close parentheses space minus 1 half space straight e to the power of straight x space cot space straight x over 2 plus straight C end style

Solution 20

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
straight I equals integral straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx
straight I equals integral straight e to the power of straight x open parentheses fraction numerator 1 over denominator 1 minus cosx end fraction minus fraction numerator sinx over denominator 1 minus cosx end fraction close parentheses dx
straight I equals integral straight e to the power of straight x open parentheses fraction numerator 1 over denominator 2 sin squared begin display style straight x over 2 end style end fraction minus fraction numerator 2 sin begin display style straight x over 2 end style cos straight x over 2 over denominator 2 sin squared straight x over 2 end fraction close parentheses dx
straight I equals integral straight e to the power of straight x open parentheses fraction numerator cosec squared straight x over 2 over denominator 2 end fraction minus cot straight x over 2 close parentheses dx
straight f left parenthesis straight x right parenthesis equals negative cot straight x over 2 rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals fraction numerator cosec squared straight x over 2 over denominator 2 end fraction
straight I equals negative straight e to the power of straight x cot straight x over 2 plus straight C end style

Question 21

begin mathsize 12px style integral 2 over open parentheses straight e to the power of straight x space plus space straight e to the power of negative straight x end exponent close parentheses squared dx
open parentheses straight a close parentheses space fraction numerator negative straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction plus straight C
open parentheses straight b close parentheses space minus fraction numerator begin display style 1 end style over denominator begin display style straight e to the power of straight x plus straight e to the power of negative straight x end exponent end style end fraction plus straight C
open parentheses straight c close parentheses space fraction numerator begin display style negative 1 end style over denominator begin display style open parentheses straight e to the power of straight x plus space 1 close parentheses squared end style end fraction plus straight C
open parentheses straight d close parentheses space fraction numerator begin display style 1 end style over denominator begin display style straight e to the power of straight x plus straight e to the power of negative straight x end exponent end style end fraction plus straight C end style

Solution 21

Error converting from MathML to accessible text.

Question 22

begin mathsize 12px style integral fraction numerator straight e to the power of straight x open parentheses 1 plus straight x close parentheses over denominator cos squared open parentheses xe to the power of straight x close parentheses end fraction dx equals

open parentheses straight a close parentheses space 2 space log subscript straight e cos open parentheses xe to the power of straight x close parentheses plus straight C
open parentheses straight b close parentheses space sec open parentheses xe to the power of straight x close parentheses plus straight C
open parentheses straight c close parentheses space tan open parentheses xe to the power of straight x close parentheses plus straight C
open parentheses straight d close parentheses space tan open parentheses straight x plus straight e to the power of straight x close parentheses plus straight C end style

Solution 22

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
straight I equals integral fraction numerator straight e to the power of straight x open parentheses 1 plus straight x close parentheses over denominator cos squared open parentheses xe to the power of straight x close parentheses end fraction dx
Put space xe to the power of straight x equals straight t
rightwards double arrow straight e to the power of straight x open parentheses 1 plus straight x close parentheses dx equals dt
straight I equals integral fraction numerator dt over denominator cos squared straight t end fraction
straight I equals integral sec squared tdt
straight I equals tant plus straight C
straight I equals tan open parentheses xe to the power of straight x close parentheses plus straight C end style


Question 23

begin mathsize 12px style integral fraction numerator sin squared straight x over denominator cos to the power of 4 straight x end fraction dx equals

open parentheses straight a close parentheses space 1 third space tan squared straight x space plus space straight C
open parentheses straight b close parentheses space 1 half space tan squared straight x space plus space straight C
open parentheses straight c close parentheses space 1 third space tan cubed straight x space plus space straight C
open parentheses straight d close parentheses space none space of space these end style

Solution 23

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
straight I equals integral fraction numerator sin squared straight x over denominator cos to the power of 4 straight x end fraction dx
straight I equals integral tan squared xsec squared xdx
Put space tanx equals straight t rightwards double arrow sec squared xdx equals dt
straight I equals integral straight t squared dt
straight I equals straight t cubed over 3 plus straight C
straight I equals fraction numerator tan cubed straight x over denominator 3 end fraction plus straight C end style

Question 24

begin mathsize 12px style The space primitive space of space the space function space straight f left parenthesis straight x right parenthesis space equals space open parentheses 1 minus 1 over straight x squared close parentheses straight a to the power of straight x plus 1 over straight x end exponent comma space straight a space greater than space 0 space is
open parentheses straight a close parentheses space fraction numerator straight a to the power of straight x plus begin display style 1 over straight x end style end exponent over denominator log subscript straight e straight a end fraction
open parentheses straight b close parentheses space log subscript straight e straight a times straight a to the power of straight x plus 1 over straight x end exponent
open parentheses straight c close parentheses space straight a to the power of straight x plus begin display style 1 over straight x end style end exponent over straight x log subscript straight e straight a
open parentheses straight d close parentheses space straight x fraction numerator straight a to the power of straight x plus begin display style 1 over straight x end style end exponent over denominator log subscript straight e straight a end fraction end style

Solution 24

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
straight f left parenthesis straight x right parenthesis equals open parentheses 1 minus 1 over straight x squared close parentheses straight a to the power of straight x plus 1 over straight x end exponent
rightwards double arrow integral straight f left parenthesis straight x right parenthesis dx equals integral open parentheses 1 minus 1 over straight x squared close parentheses straight a to the power of straight x plus 1 over straight x end exponent dx
Put space straight x plus 1 over straight x equals straight t
rightwards double arrow open parentheses 1 minus 1 over straight x squared close parentheses dx equals dt
straight I equals integral straight a to the power of straight t dt
straight I equals fraction numerator straight a to the power of straight t over denominator log subscript straight e straight a end fraction plus straight C
straight I equals fraction numerator straight a to the power of straight x plus 1 over straight x end exponent over denominator log subscript straight e straight a end fraction plus straight C end style

Question 25

begin mathsize 12px style The space value space of space integral fraction numerator 1 over denominator straight x plus straight x space log space straight x end fraction dx space is
open parentheses straight a close parentheses space 1 space plus space log space straight x
open parentheses straight b close parentheses space straight x space plus space log space straight x
open parentheses straight c close parentheses space straight x space log space open parentheses 1 space plus space log space straight x close parentheses
open parentheses straight d close parentheses space log open parentheses 1 space plus space log space straight x close parentheses end style

Solution 25

begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis
straight I equals integral fraction numerator 1 over denominator straight x plus xlogx end fraction dx
straight I equals integral fraction numerator dx over denominator straight x open parentheses 1 plus logx close parentheses end fraction
Put space 1 plus logx equals straight t
rightwards double arrow 1 over straight x dx equals dt
straight I equals integral 1 over straight t dt
rightwards double arrow straight I equals log open vertical bar straight t close vertical bar plus straight C
straight I equals log open parentheses 1 plus logx close parentheses plus straight C
end style

Question 26

begin mathsize 12px style integral square root of fraction numerator straight x over denominator 1 minus straight x end fraction end root dx space is space equal space to
open parentheses straight a close parentheses space sin to the power of negative 1 end exponent square root of straight x plus straight C
open parentheses straight b close parentheses space sin to the power of negative 1 end exponent open curly brackets square root of straight x minus square root of straight x left parenthesis 1 minus straight x right parenthesis end root close curly brackets plus straight C
open parentheses straight c close parentheses space sin to the power of negative 1 end exponent open curly brackets square root of straight x left parenthesis 1 minus straight x right parenthesis end root close curly brackets plus straight C
open parentheses straight d close parentheses space sin to the power of negative 1 end exponent square root of straight x minus square root of straight x left parenthesis 1 minus straight x right parenthesis end root plus straight C end style

Solution 26

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
straight I equals integral square root of fraction numerator straight x over denominator 1 minus straight x end fraction end root dx
straight I equals integral square root of fraction numerator straight x over denominator 1 minus straight x end fraction cross times straight x over straight x end root dx
straight I equals integral fraction numerator xdx over denominator square root of straight x minus straight x squared end root end fraction
Consider comma
straight x equals straight A fraction numerator straight d open parentheses straight x minus straight x squared close parentheses over denominator dx end fraction plus straight B
straight x equals straight A open parentheses 1 minus 2 straight x close parentheses plus straight B
straight x equals negative 2 Ax plus straight A plus straight B
minus 2 straight A equals 1 rightwards double arrow straight A equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow straight A plus straight B equals 0 rightwards double arrow straight B equals 1 half
straight I equals integral fraction numerator fraction numerator negative 1 over denominator 2 end fraction open parentheses 1 minus 2 straight x close parentheses plus 1 half over denominator square root of straight x minus straight x squared end root end fraction dx
straight I equals integral open parentheses fraction numerator negative 1 over denominator 2 end fraction fraction numerator 1 minus 2 straight x over denominator square root of straight x minus straight x squared end root end fraction plus fraction numerator 1 over denominator 2 square root of straight x minus straight x squared end root end fraction close parentheses dx
straight I equals fraction numerator negative 1 over denominator 2 end fraction cross times 2 square root of straight x minus straight x squared end root plus 1 half integral fraction numerator 1 over denominator square root of straight x minus straight x squared end root end fraction dx
Second space term space after space completing space square space method space you space will space get space as
straight I equals negative square root of straight x minus straight x squared end root plus sin to the power of negative 1 end exponent square root of straight x plus straight C end style

Question 27

begin mathsize 12px style integral straight e to the power of straight x open curly brackets straight f open parentheses straight x close parentheses space plus space straight f apostrophe open parentheses straight x close parentheses close curly brackets dx space equals

open parentheses straight a close parentheses space straight e to the power of straight x space straight f open parentheses straight x close parentheses space plus space straight C
open parentheses straight b close parentheses space straight e to the power of straight x space plus space straight f open parentheses straight x close parentheses space plus space straight C
open parentheses straight c close parentheses space 2 straight e to the power of straight x space straight f open parentheses straight x close parentheses space plus space straight C
open parentheses straight d close parentheses space straight e to the power of straight x space minus space straight f open parentheses straight x close parentheses space plus space straight C end style

Solution 27

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
integral straight e to the power of straight x open curly brackets straight f open parentheses straight x close parentheses plus straight f apostrophe open parentheses straight x close parentheses close curly brackets dx equals straight e to the power of straight x straight f open parentheses straight x close parentheses plus straight C
end style

Question 28

begin mathsize 12px style The space value space of space integral fraction numerator sin space straight x space plus space cos space straight x over denominator square root of 1 minus sin space 2 straight x end root end fraction dx space is space equal space to
open parentheses straight a close parentheses space square root of sin space 2 straight x space end root plus space straight C
open parentheses straight b close parentheses space square root of cos 2 straight x end root space plus space straight C
open parentheses straight c close parentheses space plus-or-minus open parentheses sin space straight x space minus space cos space straight x close parentheses space plus space straight C
open parentheses straight d close parentheses space plus-or-minus space log space open parentheses sin space straight x space minus space cos space straight x close parentheses space plus space straight C end style

Solution 28

begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis
straight I equals integral fraction numerator sinx plus cosx over denominator square root of 1 minus sin 2 straight x end root end fraction dx
straight I equals integral fraction numerator sinx plus cosx over denominator cosx minus sinx end fraction dx
Put space cosx minus sinx equals straight t rightwards double arrow open parentheses sinx plus cosx close parentheses dx equals plus-or-minus dt
rightwards double arrow open parentheses sinx plus cosx close parentheses dx equals plus-or-minus dt
straight I equals plus-or-minus integral dt over straight t
straight I equals plus-or-minus log open vertical bar straight t close vertical bar plus straight C
straight I equals plus-or-minus log open vertical bar sinx minus cosx close vertical bar plus straight C
end style

Question 29

begin mathsize 12px style If space integral straight x space sin space straight x space dx space equals space minus straight x space cos space straight x space plus space straight alpha comma space then space straight alpha space is space equal space to

open parentheses straight a close parentheses space sin space straight x space plus space straight C
open parentheses straight b close parentheses space cos space straight x space plus space straight C
open parentheses straight c close parentheses space straight C
open parentheses straight d close parentheses space none space fo space these end style

Solution 29

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
integral xsinxdx equals negative xcosx plus straight alpha
straight I equals integral xsinxdx
straight I equals straight x integral sinxdx minus integral open parentheses fraction numerator d straight x over denominator d straight x end fraction integral sinxdx close parentheses dx
straight I equals negative xcosx plus integral cosxdx
straight I equals xcosx plus sinx plus straight C
rightwards double arrow straight alpha equals sinx plus straight C
end style

Question 30

begin mathsize 12px style integral fraction numerator cos space 2 straight x space minus space 1 over denominator cos space 2 straight x space plus space 1 end fraction dx equals

open parentheses straight a close parentheses space tan space straight x space minus space straight x space plus space straight C
open parentheses straight b close parentheses space straight x space plus space tan space straight x space plus space straight C
open parentheses straight c close parentheses space straight x space minus space tan space straight x space plus space straight C
open parentheses straight d close parentheses space minus straight x space minus space cot space straight x space plus space straight C end style

Solution 30

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
straight I equals integral fraction numerator cos 2 straight x minus 1 over denominator cos 2 straight x plus 1 end fraction dx
straight I equals negative integral fraction numerator 1 minus cos 2 straight x over denominator 1 plus cos 2 straight x end fraction dx
straight I equals negative integral fraction numerator 2 sin squared straight x over denominator 2 cos squared straight x over 2 end fraction dx
straight I equals negative integral tan squared xdx
straight I equals negative integral open parentheses sec squared straight x minus 1 close parentheses dx
straight I equals negative open parentheses tanx minus straight x close parentheses plus straight C
straight I equals straight x minus tanx plus straight C end style

Question 31

begin mathsize 12px style integral fraction numerator cos space 2 straight x space minus space cos space 2 straight theta over denominator cos space straight x space minus space cos space straight theta end fraction dx space is space equal space to

open parentheses straight a close parentheses space 2 open parentheses sin space straight x space plus space straight x space cos space straight theta close parentheses space plus space straight C
open parentheses straight b close parentheses space 2 open parentheses sin space straight x space minus space straight x space cos space straight theta close parentheses space plus space straight C
open parentheses straight c close parentheses space 2 open parentheses sin space straight x space plus space 2 straight x space cos space straight theta close parentheses space plus space straight C
open parentheses straight d close parentheses space 2 open parentheses sin space straight x space minus space 2 straight x space cos space straight theta close parentheses space plus space straight C end style

Solution 31

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
straight I equals integral fraction numerator cos 2 straight x minus cos 2 straight theta over denominator cosx minus cosθ end fraction dx
straight I equals integral fraction numerator 2 cos squared straight x minus 1 minus open parentheses 2 cos squared straight theta minus 1 close parentheses over denominator cosx minus cosθ end fraction dx
straight I equals integral fraction numerator 2 open parentheses cos squared straight x minus cos squared straight theta close parentheses over denominator cosx minus cosθ end fraction dx
straight I equals 2 integral open parentheses cosx plus cosθ close parentheses dx
straight I equals 2 open parentheses sinx plus xcosθ close parentheses plus straight C end style

Question 32

begin mathsize 12px style integral straight x to the power of 9 over open parentheses 4 straight x squared plus 1 close parentheses to the power of 6 dx space is space equal space to

open parentheses straight a close parentheses space fraction numerator 1 over denominator 5 straight x end fraction open parentheses 4 plus 1 over straight x squared close parentheses to the power of negative 5 end exponent plus straight C
open parentheses straight b close parentheses space fraction numerator begin display style 1 end style over denominator begin display style 5 end style end fraction open parentheses 4 plus fraction numerator begin display style 1 end style over denominator begin display style straight x squared end style end fraction close parentheses to the power of negative 5 end exponent plus straight C
open parentheses straight c close parentheses space fraction numerator begin display style 1 end style over denominator begin display style 10 straight x end style end fraction open parentheses fraction numerator begin display style 1 end style over denominator begin display style straight x squared end style end fraction plus 4 close parentheses to the power of negative 5 end exponent plus straight C
open parentheses straight d close parentheses space fraction numerator begin display style 1 end style over denominator begin display style 10 end style end fraction open parentheses fraction numerator begin display style 1 end style over denominator begin display style straight x squared end style end fraction plus 4 close parentheses to the power of negative 5 end exponent plus straight C
end style

Solution 32

begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis
straight I equals integral straight x to the power of 9 over open parentheses 4 straight x squared plus 1 close parentheses to the power of 6 dx
straight I equals integral fraction numerator straight x to the power of 9 over denominator straight x to the power of 12 open parentheses 4 plus begin display style 1 over straight x squared end style close parentheses to the power of 6 end fraction dx
straight I equals integral fraction numerator dx over denominator straight x cubed open parentheses 4 plus 1 over straight x squared close parentheses to the power of 6 end fraction
Put space open parentheses 4 plus 1 over straight x squared close parentheses equals straight t rightwards double arrow fraction numerator negative 2 over denominator straight x cubed end fraction dx equals dt
straight I equals 1 half integral negative straight t to the power of negative 6 end exponent dt
straight I equals 1 half open parentheses fraction numerator negative straight t to the power of negative 5 end exponent over denominator negative 5 end fraction close parentheses plus straight C
straight I equals 1 over 10 1 over straight t to the power of 5 plus straight C
straight I equals 1 over 10 open parentheses 4 plus 1 over straight x squared close parentheses to the power of negative 5 end exponent plus straight C end style

Question 33

begin mathsize 12px style integral fraction numerator straight x cubed over denominator square root of 1 plus straight x squared end root end fraction dx equals straight a open parentheses 1 plus straight x squared close parentheses to the power of 3 divided by 2 end exponent space plus space straight b square root of 1 plus straight x squared end root plus straight C comma space then

open parentheses straight a close parentheses space straight a space equals space 1 third comma space straight b space equals space 1
open parentheses straight b close parentheses space straight a space equals space minus fraction numerator begin display style 1 end style over denominator begin display style 3 end style end fraction comma space straight b space equals space 1
open parentheses straight c close parentheses space straight a space equals negative fraction numerator begin display style 1 end style over denominator begin display style 3 end style end fraction comma space straight b space equals space minus 1
open parentheses straight d close parentheses space straight a space equals space fraction numerator begin display style 1 end style over denominator begin display style 3 end style end fraction comma space straight b space equals space minus 1 end style

Solution 33

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
straight I equals integral fraction numerator straight x cubed over denominator square root of 1 plus straight x squared end root end fraction dx
1 plus straight x squared equals straight t rightwards double arrow 2 xdx equals dt
rightwards double arrow xdx equals dt over 2
straight I equals integral fraction numerator straight x squared over denominator square root of 1 plus straight x squared end root end fraction xdx
straight I equals integral fraction numerator straight t minus 1 over denominator square root of straight t end fraction dt over 2
straight I equals 1 half integral open parentheses square root of straight t minus straight t to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent close parentheses dt
straight I equals 1 half open parentheses 2 over 3 straight t to the power of 3 over 2 end exponent minus 2 square root of straight t close parentheses plus straight C
straight I equals 1 third open parentheses 1 plus straight x squared close parentheses to the power of 3 over 2 end exponent minus square root of 1 plus straight x squared end root plus straight C
straight a equals 1 third comma space straight b equals negative 1 end style

Question 34

begin mathsize 12px style integral fraction numerator straight x cubed over denominator straight x plus 1 end fraction dx space is space equal space to

open parentheses straight a close parentheses space straight x space plus space straight x squared over 2 plus straight x cubed over 3 minus space log space open vertical bar 1 minus straight x close vertical bar plus straight C
open parentheses straight b close parentheses space straight x space plus space fraction numerator begin display style straight x squared end style over denominator begin display style 2 end style end fraction minus fraction numerator begin display style straight x cubed end style over denominator begin display style 3 end style end fraction minus space log space open vertical bar 1 minus straight x close vertical bar plus straight C
open parentheses straight c close parentheses space straight x space minus space fraction numerator begin display style straight x squared end style over denominator begin display style 2 end style end fraction minus fraction numerator begin display style straight x cubed end style over denominator begin display style 3 end style end fraction minus space log space open vertical bar 1 plus straight x close vertical bar plus straight C
open parentheses straight d close parentheses space straight x space minus space fraction numerator begin display style straight x squared end style over denominator begin display style 2 end style end fraction plus fraction numerator begin display style straight x cubed end style over denominator begin display style 3 end style end fraction minus space log space open vertical bar 1 plus straight x close vertical bar plus straight C
end style

Solution 34

begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis
straight I equals integral fraction numerator straight x cubed over denominator straight x plus 1 end fraction dx
straight I equals integral fraction numerator straight x cubed plus 1 minus 1 over denominator straight x plus 1 end fraction dx
straight I equals integral fraction numerator open parentheses straight x plus 1 close parentheses open parentheses straight x squared minus straight x plus 1 close parentheses minus 1 over denominator straight x plus 1 end fraction dx
straight I equals integral open parentheses straight x squared minus straight x plus 1 minus fraction numerator 1 over denominator straight x plus 1 end fraction close parentheses dx
straight I equals straight x cubed over 3 minus straight x squared over 2 plus straight x minus log open vertical bar straight x plus 1 close vertical bar plus straight C end style

Question 35

begin mathsize 12px style If space integral fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction dx space equals space straight a space log space open vertical bar 1 plus straight x squared close vertical bar space plus space straight b space tan to the power of negative 1 end exponent straight x space plus space 1 fifth log open vertical bar straight x space plus space 2 close vertical bar space plus space straight C comma space then

open parentheses straight a close parentheses space straight a space equals space minus 1 over 10 comma space straight b space equals space minus 2 over 5
open parentheses straight b close parentheses space straight a space equals space 1 over 10 comma space straight b space equals space minus 2 over 5
open parentheses straight c close parentheses space straight a space equals space minus 1 over 10 comma space straight b space equals space 2 over 5
open parentheses straight d close parentheses space straight a space equals space 1 over 10 comma space straight b space equals space 2 over 5 end style

Solution 35

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
straight I equals integral fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction dx
Consider comma
fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction equals fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator open parentheses straight x squared plus 1 close parentheses end fraction
1 equals straight A open parentheses straight x squared plus 1 close parentheses plus open parentheses Bx plus straight C close parentheses open parentheses straight x plus 2 close parentheses
Comaring space coefficeints space and space solving space it space simultaneously space we space get
straight A equals 1 fifth comma space straight B equals negative 1 fifth comma space straight C equals 2 over 5
straight I equals integral open parentheses fraction numerator 1 over denominator 5 straight x plus 2 end fraction plus fraction numerator begin display style fraction numerator negative 1 over denominator 5 end fraction end style straight x plus begin display style 2 over 5 end style over denominator straight x squared plus 1 end fraction close parentheses dx
Integrating space we space get space as
1 fifth log open vertical bar straight x plus 2 close vertical bar minus 1 over 10 log open vertical bar straight x squared plus 1 close vertical bar plus 2 over 5 tan to the power of negative 1 end exponent straight x plus straight C
rightwards double arrow straight a equals negative 1 over 10 comma space straight b equals 2 over 5 end stylebegin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
straight I equals integral fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction dx
Consider comma
fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction equals fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator open parentheses straight x squared plus 1 close parentheses end fraction
1 equals straight A open parentheses straight x squared plus 1 close parentheses plus open parentheses Bx plus straight C close parentheses open parentheses straight x plus 2 close parentheses
Comaring space coefficeints space and space solving space it space simultaneously space we space get
straight A equals 1 fifth comma space straight B equals negative 1 fifth comma space straight C equals 2 over 5
straight I equals integral open parentheses fraction numerator 1 over denominator 5 straight x plus 2 end fraction plus fraction numerator begin display style fraction numerator negative 1 over denominator 5 end fraction end style straight x plus begin display style 2 over 5 end style over denominator straight x squared plus 1 end fraction close parentheses dx
Integrating space we space get space as
1 fifth log open vertical bar straight x plus 2 close vertical bar minus 1 over 10 log open vertical bar straight x squared plus 1 close vertical bar plus 2 over 5 tan to the power of negative 1 end exponent straight x plus straight C
rightwards double arrow straight a equals negative 1 over 10 comma space straight b equals 2 over 5 end style

Chapter 19 - Indefinite Integrals Excercise Ex. 19VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60