# RD Sharma Solution CLass 12 Mathematics Chapter 20 Definite Integrals

## 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

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

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

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

Evaluate the following integrals:

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

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

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

Evaluate the integral

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

For

Using
Integration By parts

For

Using
Integration By parts

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

[x]=0 for 0 < x

and [x]=1 for 1< x < 2

Hence

Question 30

Solution 30

## 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

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

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

Solution 34

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

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 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

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. 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

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

## 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)

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

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)

Question 34

(a) p/4

(b) p/2

(c) p

(d) 1

Solution 34

Correct option:(d)

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)

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)

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