RD SHARMA Solutions for Class 12-science Maths Chapter 20 - Definite Integrals

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

On TopperLearning, you can find study resources such as sample papers, mock tests, Class 12 Maths NCERT solutions and more. These learning materials can help you understand concepts such as differentiation of functions, direction cosines, integrals, and more. Also, you can practise the Maths problems by going through the solutions given by our experts.

Maths is considered as one of the most difficult subjects in CBSE Class 12 Science. Our Maths experts simplify complex Maths problems by assisting you with the right methods to solve problems and score full marks. You may still have doubts while referring to the Maths revision notes or Maths NCERT solutions. Solve those doubts by asking an expert through the “Undoubt” feature on the student dashboard.

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Chapter 20 - Definite Integrals Exercise Ex. 20.1

Question 1
Solution 1
Question 2

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

Solution 5

T h e r e f o r e comma integral subscript 2 superscript 3 fraction numerator x over denominator x squared plus 1 end fraction equals 1 half log 2

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

T h e r e f o r e comma space I equals 2 to the power of begin display style 5 over 2 end style end exponent over 3

Question 54
Solution 54
Question 55

  

Solution 55

 

  

  

 

Let cosx =u , Then

  

  

Hence

  

  

  

Question 56
Solution 56
Question 57

Solution 57

Question 58

  

Solution 58

  

Question 59

  

Solution 59

  

  

  

  

  

Given :

  

  

  

  

Question 60

Solution 60

Question 61

  

Solution 61

  

  

  

  

 

Question 62

  

Solution 62

Question 63

  

Solution 63

 

  

  

  

  

  

We know , By reduction formula 

  

For n=2

  

  

For n=4

  

  

Hence

  

  

 

Note: Answer given at back is incorrect.

Question 64

  

Solution 64

Using Integration By parts

  

  

 

 

  

 

Question 65

  

Solution 65

  

  

  

  

  

  

  

  

Question 66

  

Solution 66

 

Note: Answer given in the book is incorrect. 

Question 67

  

Solution 67

 =(1/4)log(2e)

 

Chapter 20 - Definite Integrals Exercise Ex. 20.2

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

Using Integration By parts

  

  

  

  

  

Hence

  

  

  

 

Question 25

  

Solution 25

  

Question 26
Solution 26
Question 27

Evaluate begin mathsize 11px style integral subscript 0 superscript straight pi fraction numerator 1 over denominator 5 plus 3 space cos space straight x end fraction dx end style

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

  

Solution 60

  

  

  

 

Question 61

  

Solution 61

  

 

Question 62

  

Solution 62

  

Chapter 20 - Definite Integrals Exercise Ex. 20.3

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

2x+3 is positive for x>-1.5 . Hence

  

  

  

  

  

 

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 integral integral subscript 1 superscript 4 open curly brackets open vertical bar x minus 1 close vertical bar plus open vertical bar x minus 2 close vertical bar plus open vertical bar x minus 4 close vertical bar close curly brackets d x

Solution 19

L e t space I equals integral subscript 1 superscript 4 open curly brackets open vertical bar x minus 1 close vertical bar plus open vertical bar x minus 2 close vertical bar plus open vertical bar x minus 4 close vertical bar close curly brackets d x
equals integral subscript 1 superscript 2 open curly brackets open parentheses x minus 1 close parentheses minus open parentheses x minus 2 close parentheses minus open parentheses x minus 4 close parentheses close curly brackets d x plus integral subscript 2 superscript 4 open curly brackets open parentheses x minus 1 close parentheses plus open parentheses x minus 2 close parentheses minus open parentheses x minus 4 close parentheses close curly brackets d x
equals integral subscript 1 superscript 2 open curly brackets open parentheses x minus 1 minus x plus 2 minus x plus 4 close parentheses close curly brackets d x plus integral subscript 2 superscript 4 open curly brackets open parentheses x minus 1 plus x minus 2 minus x plus 4 close parentheses close curly brackets d x
equals integral subscript 1 superscript 2 open parentheses 5 minus x close parentheses d x plus integral subscript 2 superscript 4 open parentheses x plus 1 close parentheses d x
equals open square brackets 5 x minus x squared over 2 close square brackets table row 2 row 1 end table plus open square brackets x squared over 2 plus x close square brackets table row 4 row 2 end table
equals open square brackets 10 minus 2 minus 5 plus 1 half close square brackets plus open square brackets 8 plus 4 minus 2 minus 2 close square brackets
equals 7 over 2 plus 8
I equals 23 over 2

Question 20

Solution 20

Question 21

Solution 21

Question 22

  

Solution 22

  

  

  

  

  

Question 23

  

Solution 23

  

For

  

Using Integration By parts

  

  

  

  

  

For

  

Using Integration By parts

  

  

  

  

Question 24

  

Solution 24

  

  

 

Question 25

  

Solution 25

  

Question 26

  

Solution 26

Question 27

  

Solution 27

  

  

Question 28

  

Solution 28

[x]=0 for 0

and [x]=1 for 1

Hence

Question 29

  

Solution 29

  

Question 30

Evaluate the following integrals:

begin mathsize 12px style integral from negative straight pi divided by 2 to straight pi divided by 2 of fraction numerator negative straight pi divided by 2 over denominator square root of cosx space sin squared straight x end root end fraction dx end style


Solution 30

NOTE: Answer not matching with back answer.

Chapter 20 - Definite Integrals Exercise Ex. 20.4A

Question 1

  

Solution 1

We know

  

Hence

  

We know

  

  

If

  

Then also

  

Hence

  

Question 2

  

Solution 2

We know

  

Hence

  

If

  

Then

  

Question 3

  

Solution 3

We know

  

Hence

  

If

  

Then

  

So

  

 

Question 4

  

Solution 4

We know

  

Hence

  

If

  

Then

  

Hence

Question 5

  

Solution 5

We know

  

Hence

  

If

  

Then

  

So

  

We know

 

If f(x) is even

  

If f(x) is odd

  

Here

  

f(x) is even, hence

  

 

Note: Answer given in the book is incorrect.

Question 6

  

Solution 6

We know

  

Hence

  

If

  

Then

  

So

Question 7

  

Solution 7

We know

  

Hence

  

If

  

Then

  

So

Question 8

  

Solution 8

We know

  

Hence

  

If

  

Then

  

 

So

 

Note: Answer given in the book is incorrect. 

Question 9

  

Solution 9

  

If f(x) is even

  

If f(x) is odd

  

Here

  is odd and

  is even. Hence

  

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 20 - Definite Integrals Exercise Ex. 20.4B

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

B

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Error: the service is unavailable.

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

  

  

  

Hence

  

Question 20

  

Solution 20

  

Question 21

  

Solution 21

  

Now

  

Let cosx=t

  

  

  

  

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

Evaluate the integral integral subscript 0 superscript 1 log open parentheses 1 over x minus 1 close parentheses d x

Solution 34

L e t space I equals integral subscript 0 superscript 1 log open parentheses 1 over x minus 1 close parentheses d x
equals integral subscript 0 superscript 1 log open parentheses fraction numerator 1 minus x over denominator x end fraction close parentheses d x
equals integral subscript 0 superscript 1 log open parentheses 1 minus x close parentheses d x minus integral subscript 0 superscript 1 log open parentheses x close parentheses d x
A p p l y i n g space t h e space p r o p e r t y comma space integral subscript 0 superscript a f open parentheses x close parentheses d x equals integral subscript 0 superscript a f open parentheses a minus x close parentheses d x
T h u s comma space I equals integral subscript 0 superscript 1 log open parentheses 1 minus open parentheses 1 minus x close parentheses close parentheses d x minus integral subscript 0 superscript 1 log open parentheses x close parentheses d x
equals integral subscript 0 superscript 1 log open parentheses 1 minus 1 plus x close parentheses d x minus integral subscript 0 superscript 1 log open parentheses x close parentheses d x
equals integral subscript 0 superscript 1 log open parentheses x close parentheses d x minus integral subscript 0 superscript 1 log open parentheses x close parentheses d x
equals 0

Question 35

  

Solution 35

  

Question 36

  

Solution 36

  

  

  

  

Question 37

  

Solution 37

  

Question 38

  

Solution 38

We know

  

Also here

  

So

  

  

Hence

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

  

Chapter 20 - Definite Integrals Exercise Ex. 20RE

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

Evaluate the integral integral subscript 0 superscript pi over 2 end superscript x squared cos 2 x d x

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

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

Chapter 20 - Definite Integrals Exercise Ex. 20.5

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

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

Chapter 20 - Definite Integrals Exercise MCQ

Question 1

Mark the correct alternative in each of the following:

(a) p/2

(b) p/4

(c) p/6

(d) p/8

Solution 1

Correct option: (d)

  

Question 2

(a) 0

(b) 1/2

(c) 2

(d) 3/2

Solution 2

Correct option: (c)

  

Question 3

 

Solution 3

Correct option: (a)

  

Question 4

(a) 0

(b) 2

(c) 8

(d) 4

 

Solution 4

Correct option: (c)

  

Note: Answer not matching with back answer.

Question 5

(a) 0

(b) p/2

(c) p/4

(d) None of these

 

Solution 5

Correct option:(c)

 

Question 6

(a) Log 2-1

(b) Log 2

(c) Log 4-1

(d) -log 2

 

Solution 6

Correct option: (b)

  

Question 7

(a) 2

(b) 1

(c) p/4

(d) p2/8

 

Solution 7

Correct option: (a)

  

Question 8

 

Solution 8

Correct option: (d)

  

 

Question 9

 

Solution 9

Correct option: (b)

  

Question 10

 

Solution 10

Note: Answer not matching with back answer.

Question 11

 

 

Solution 11

Correct option: (a)

  

Question 12

(a) p/3

(b) p/6

(c) p/12

(d) p/2

 

Solution 12

Correct option:(c)

  

 

Question 13

 

Solution 13

Correct option: (a)

  

Question 14

(a) 1

(b) e-1

(c) e+1

(d) 0

 

Solution 14

Correct option: (a)

  

 

Question 15

 

Solution 15

Correct option:(a)

  

Question 16

 

Solution 16

Correct option:(a)

  

 

Question 17

 

Solution 17

Correct option:(b)

  

Question 18

(a) 1

(b) 2

(c) -1

(d) -2

 

Solution 18

Correct option: (b)

  

Question 19

 

Solution 19

Correct option: (a)

  

Question 20

(a) 1

(b) e-1

(c) 0

(d) -1

 

Solution 20

Correct option: (b)

  

 

Question 21

 

Solution 21

Correct option:(b)

  

 

Question 22

(a) 4a2 

(b) 0

(c) 2a2 

(d) None of these

 

Solution 22

Correct option: (b)

  

 

Question 23

 

Solution 23

Correct option: (c)

  

Question 24

 

Solution 24

Correct option: (b)

  

 

Question 25

(a) -2

(b) 2

(c) 0

(d) 4

 

Solution 25

Correct option: (b)

  

Question 26

 

Solution 26

Correct option:(c)

  

Question 27

 

Solution 27

Correct option: (b)

  

Question 28

 

Solution 28

Correct option: (d)

 

Note: Question is modified.

 

Question 29

Solution 29

Correct option: (c)

  

Question 30

(a) 4

(b) 2

(c) -2

(d) 0

 

Solution 30

Correct option:(a)

  

Question 31

(a) 0

(b) 1

(c) p/2

(d) p/4

 

Solution 31

Correct option:(d)

  

 

Question 32

(a) p 

(b) p/2

(c) p/3

(d) p/4

 

Solution 32

Correct option: (d)

  

 

Question 33

(a) 0

(b) p 

(c) p/2

(d) p/4

 

Solution 33

Correct option:(c)

  

Note: Answer not matching with back answer.

Question 34

(a) p/4

(b) p/2

(c) p 

(d) 1

 

Solution 34

Correct option:(d)

  

Note: Answer not matching with back answer.

Question 35

(a) p 

(b) p/2

(c) 0

(d) 2p 

 

Solution 35

Correct option: (c)

  

 

Question 36

(a) p/4

(b) p/8

(c) p/2

(d) 0

 

Solution 36

Correct option: (a)

  

Question 37

(a) p In 2

(b) -p In 2

(c) 0

(d) 

 

Solution 37

Correct option:(d)

  

NOTE: Answer is not matching with back answer. 

 

Question 38

 

Solution 38

Correct option: (c)

  

Question 39

 

Solution 39

Correct option: (d)

Question 40

(a) 1

(b) 0

(c) -1

(d) p/4

 

Solution 40

Correct option: (b)

  

Question 41

(a) 2

(b) 

(c) 0

(d) -2

Solution 41

Correct option: (c)

  

 

Question 42

 

(a) 0

(b) 2

(c) p 

(d) 1

Solution 42

Correct option: (c)

  

Chapter 20 - Definite Integrals Exercise Ex. 20VSAQ

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

G i v e n space t h a t space integral subscript 0 superscript a 3 x squared d x equals 8
rightwards double arrow open square brackets fraction numerator 3 x cubed over denominator 3 end fraction close square brackets subscript 0 superscript a equals 8
rightwards double arrow a cubed minus 0 equals 8
rightwards double arrow a cubed equals 8
rightwards double arrow a equals 2

Question 38

Solution 38

Question 39

Solution 39

Question 40

Note: The lower limit is incorrect in textbook. Consider the lower limit as '0'.

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

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