RS Aggarwal Solutions for Class 12 Maths Chapter 12: Indefinite Integral

We know that the inverse process of differentiation is integration. This exercise covers the concept of integration and important formulas, which are used in solving the problems. The solutions are designed by highly experienced faculty in a stepwise manner as per the exam pattern of the current board. This improves the analytical abilities among students which help them in their exam preparation. In order to score good marks in the board exams, students can access RS Aggarwal Solutions for Class 12 Maths Chapter 12 Indefinite Integral free PDF, from the links which are given below.

RS Aggarwal Solutions for Class 12 Maths Chapter 12: Indefinite Integral Download PDF

 

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Access RS Aggarwal Solutions for Class 12 Maths Chapter 12: Indefinite Integral

Exercise 12 page: 597

Very-Short-Answer Questions

Evaluate:

1.

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 1

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 2

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 3

We know that

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 4

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 5

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 6

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 7

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 8

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 9

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 10

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 11

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 12

2.

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 13

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 14

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 15

By integration we get

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 16

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 17

3.

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 18

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 19

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 20

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 21

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 22

4.

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 23

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 24

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 25

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 26

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 27

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 28

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 29

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 30

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 31

Solution:

It can be written as

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 32

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 33

6.

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 34

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 35

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 36

It can be written as

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 37

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 38

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 39

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 40

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 41

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 42

Solution:

It is given that

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 43

By integrating w.r.t x

= – cosec x – tan x + x – sec x + 2 tan x + c

On further calculation

= – cosec x + tan x + x – sec x + c

9. (i) ∫sec x (sec x + tan x) dx

(ii) ∫cosec x (cosec x – cot x) dx

Solution:

(i) ∫sec x (sec x + tan x) dx

It can be written as

= ∫sec2 x + sec x tan x dx

By integrating w.r.t x

= tan x + sec x + c

(ii) ∫cosec x (cosec x – cot x) dx

It can be written as

= ∫cosec2 x – cosec x cot x dx

By integrating w.r.t x

= – cot x + cosec x + c

10.

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 44

Solution:

(i) ∫ (tan x + cot x) 2 dx

It can be written as

= ∫ (tan2 x + cot2 x + 2 tan x cot x) dx

So we get

= ∫ ([sec2 x – 1] + [cosec2 x – 1] + 2) dx

On further calculation

= ∫ (sec2 x + cosec2 x) dx

By integrating w.r.t x

= tan x – cot x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 45

So we get

= ∫ (sec2 x + 2 tan x sec x) dx

By integrating w.r.t x

= tan x + 2 sec x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 46

It can be written as

= ∫ (3 cot x cosec x + 4 cosec2 x) dx

By integrating w.r.t x

= – 3 cosec x – 4 cot x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 47

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 48

So we get

= ∫ (cosec2 x + cosec x cot x) dx

By integrating w.r.t x

= – cot x – cosec x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 49

Now multiply and divide by (1 + sin x)

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 50

So we get

= ∫ (sec2 x + sec x tan x) dx

By integrating w.r.t x

= tan x + sec x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 51

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 52

It can be written as

= ∫ (tan x sec x – tan2 x) dx

So we get

= ∫ (tan x sec x – [sec2 x – 1]) dx

By integrating w.r.t x

= sec x – tan x + x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 53

It can be written as

= ∫ (cosec2 x + cosec x cot x) dx

By integrating w.r.t x

= – cot x – cosec x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 54

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 55

On further calculation

= ∫ (cot x cosec x – cot2 x) dx

It can be written as

= ∫ (cot x cosec x – [cosec2 x – 1]) dx

By integrating w.r.t x

= – cosec x + cot x + x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 56

On further calculation

= ∫ (tan x sec x + tan2 x) dx

It can be written as

= ∫ (tan x sec x + [sec2 x – 1]) dx

By integrating w.r.t x

= sec x + tan x – x + c

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 57

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 58

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 59

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 60

Solution:

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 61

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 62

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 63

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 64

Solution:

It is given that

RS Aggarwal Solutions for Class 12 Chapter 12 Ex 12 Image 65

We can write it as

= ∫ √ (sin x + cos x) 2 dx

So we get

= ∫ (sin x + cos x) dx

By integrating w.r.t x

= – cos x + sin x + c

We get

= sin x – cos x + c

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