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RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 20 - Definite Integrals.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20.1

Solution 1

Solution 2

Solution 3

Solution 4

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

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

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

Solution 54

Solution 55

 

  

  

 

Let cosx =u , Then

  

  

Hence

  

  

  

Solution 56

Solution 57

Solution 58

  

Solution 59

  

  

  

  

  

Given :

  

  

  

  

Solution 60

Solution 61

  

  

  

  

 

Solution 62

Solution 63

 

  

  

  

  

  

We know , By reduction formula 

  

For n=2

  

  

For n=4

  

  

Hence

  

  

 

Note: Answer given at back is incorrect.

Solution 64

Using Integration By parts

  

  

 

 

  

 

Solution 65

  

  

  

  

  

  

  

  

Solution 66

 

Note: Answer given in the book is incorrect. 

Solution 67

 =(1/4)log(2e)

 

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20.2

Solution 1


Solution 2

Solution 3

Solution 4


Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17


Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Using Integration By parts

  

  

  

  

  

Hence

  

  

  

 

Solution 25

  

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

  

Solution 32


Solution 33


Solution 34

Solution 35

Solution 36

Solution 37


Solution 38


Solution 39

Solution 40

  

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46


Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

Solution 53

Solution 54

Solution 55

Solution 56

Solution 57

  

Solution 58

 

  

 

Solution 59

  

Solution 60

  

  

  

 

Solution 61

  

 

Solution 62

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20.3

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 2

Solution 3

Solution 4

Solution 5

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

  

  

  

  

  

 

Solution 6

  

  

  

  

  

  

  

 

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

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

Solution 18

Solution 19

Solution 20

  

  

  

  

  

Solution 21

  

For

  

Using Integration By parts

  

  

  

  

  

For

  

Using Integration By parts

  

  

  

  

Solution 22

  

  

 

Solution 23

  

Solution 24

Solution 25

  

  

Solution 27

[x]=0 for 0

Solution 28

  

Solution 26

NOTE: Answer not matching with back answer.

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20.4A

Solution 1

We know

  

Hence

  

We know

  

  

If

  

Then also

  

Hence

  

Solution 2

We know

  

Hence

  

If

  

Then

  

Solution 3

We know

  

Hence

  

If

  

Then

  

So

  

 

Solution 4

We know

  

Hence

  

If

  

Then

  

Hence

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.

Solution 6

We know

  

Hence

  

If

  

Then

  

So

Solution 7

We know

  

Hence

  

If

  

Then

  

So

Solution 8

We know

  

Hence

  

If

  

Then

  

 

So

 

Note: Answer given in the book is incorrect. 

Solution 9

  

If f(x) is even

  

If f(x) is odd

  

Here

  is odd and

  is even. Hence

  

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

  

 

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20.4B

Solution 1

Solution 2

Solution 3

B

Solution 4

Solution 5

Solution 6

Solution 7

Error: the service is unavailable.

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

  

Solution 19

  

  

  

Hence

  

Solution 20

  

Solution 21

  

Now

  

Let cosx=t

  

  

  

  

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

  

Solution 30

  

Solution 31

  

Solution 32

  

Solution 33

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

Solution 35

  

Solution 36

  

  

  

  

Solution 37

  

Solution 38

We know

  

Also here

  

So

  

  

Hence

Solution 39

  

Solution 40

Solution 41

Solution 42


Solution 43

Solution 44

Solution 45

Solution 46

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20RE

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

Solution 53

Solution 54

Solution 55

Solution 56

Solution 57

Solution 58

Solution 59

Solution 60

Solution 61

Solution 62

Solution 63

Solution 64

Solution 65

Solution 66

Solution 67

Solution 68

Solution 69

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise MCQ

Solution 1

Correct option: (d)

  

Solution 2

Correct option: (c)

  

Solution 3

Correct option: (a)

  

Solution 4

Correct option: (c)

  

Note: Answer not matching with back answer.

Solution 5

Correct option:(c)

 

Solution 6

Correct option: (b)

  

Solution 7

Correct option: (a)

  

Solution 8

Correct option: (d)

  

 

Solution 9

Correct option: (b)

  

Solution 10

Note: Answer not matching with back answer.

Solution 11

Correct option: (a)

  

Solution 12

Correct option:(c)

  

 

Solution 13

Correct option: (a)

  

Solution 14

Correct option: (a)

  

 

Solution 15

Correct option:(a)

  

Solution 16

Correct option:(a)

  

 

Solution 17

Correct option:(b)

  

Solution 18

Correct option: (b)

  

Solution 19

Correct option: (a)

  

Solution 20

Correct option: (b)

  

 

Solution 21

Correct option:(b)

  

 

Solution 22

Correct option: (b)

  

 

Solution 23

Correct option: (c)

  

Solution 24

Correct option: (b)

  

 

Solution 25

Correct option: (b)

  

Solution 26

Correct option:(c)

  

Solution 27

Correct option: (b)

  

Solution 28

Correct option: (d)

 

Note: Question is modified.

 

Solution 29

Correct option: (c)

  

Solution 30

Correct option:(a)

  

Solution 31

Correct option:(d)

  

 

Solution 32

Correct option: (d)

  

 

Solution 33

Correct option:(c)

  

Note: Answer not matching with back answer.

Solution 34

Correct option:(d)

  

Note: Answer not matching with back answer.

Solution 35

Correct option: (c)

  

 

Solution 36

Correct option: (a)

  

Solution 37

Correct option:(d)

  

NOTE: Answer is not matching with back answer. 

 

Solution 38

Correct option: (c)

  

Solution 39

Correct option: (d)

Solution 40

Correct option: (b)

  

Solution 41

Correct option: (c)

  

 

Solution 42

Correct option: (c)

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 20 - Definite Integrals Page/Excercise 20VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28 n

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

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

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 20 - Definite Integrals for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 20 - Definite Integrals.

Text Book Solutions

CBSE XII Science - Mathematics

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