# NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT solutions for class 12 maths chapter 7 Integrals: The word integration literally means summation. When you have to find the sum of finite numbers you can do by simply adding these numbers. But when you are finding the sum of a certain number of elements as the number of elements tends to infinity and at the same time each term becomes infinitesimally small, you can use a prosses to find its limit called integration. In this article, you will find NCERT solutions for class 12 maths chapter 7 integrals which is very helpful when you are solving NCERT textbook questions. In the differential calculus you learnt about the differentiation, defining tangent and and how to calculate the slope of the line. Integration is the inverse process of differentiation. Remember every function is not integrable which means you can't integrate every function. The function is integrable only if the function is already differentiated. In the solutions of NCERT for class 12 maths chapter 7 integrals, you will learn the different methods of integration for different types of functions. CBSE NCERT solutions for class 12 maths chapter 7 integrals are very important from the 12th CBSE board exam point of view and are also important for competitive examinations like JEE Main, VITEEE, BITSAT, etc. Here you will find all NCERT solutions from class 6 to 12 at a single place to help you to learn CBSE maths and science.

Integrals has 13 % weightage in 12 board final examination. Next chapter "applications of integrals" is also dependent on this chapter. So you should try to solve every problem of this chapter on your own. If you are not able to do, you can take the help of these CBSE NCERT solutions for class 12 maths chapter 7 integrals. In this chapter, there are 11 exercises with 227 questions and also 44 questions are there in miscellaneous exercise. Here, the NCERT solutions for class 12 maths chapter 7 integrals are solved and explained in detail to develop a grip on the topic. Here, you will learn two types of integrals: Definite integral and  Indefinite integral and also learn their properties and formulas.

 Definite Integral Indefinite Integral Definition A definite Integral has upper and lower limits if  'a' and 'b' are the limits or boundaries. The definite integral of f(x) is a number, not function. An integral without upper limit and lower limit. It is also an antiderivative. The indefinite integral of f(x) is a function not number. Expression

## 7.1 Introduction

7.2 Integration as an Inverse Process of Differentiation

7.2.1 Geometrical interpretation of indefinite integral

7.2.2 Some properties of indefinite integral

7.2.3 Comparison between differentiation and integration

7.3 Methods of Integration

7.3.1 Integration by substitution

7.3.2 Integration using trigonometric identities

7.4 Integrals of Some Particular Functions

7.5 Integration by Partial Fractions

7.6 Integration by Parts

7.7 Definite Integral

7.7.1 Definite integral as the limit of a sum

7.8 Fundamental Theorem of Calculus

7.8.1Area function

7.8.2 First fundamental theorem of integral calculus

7.8.3 Second fundamental theorem of integral calculus

7.9 Evaluation of Definite Integrals by Substitution

7.10 Some Properties of Definite Integrals

## NCERT Solutions for Class 12 Maths Chapter 7 Integrals- Exercise Questions

Solutions of NCERT for class 12 maths chapter 7 Integrals Exercise: 7.2

### Question:1 Integrate the functions

Given to integrate  function,

Let us assume

we get,

now back substituting the value of

as  is positive we can write

### Question:2 Integrate the functions

Given to integrate  function,

Let us assume

we get,

### Question:3  Integrate the functions

Given to integrate  function,

Let us assume

we get,

### Question:4 Integrate the functions

Given to integrate  function,

Let us assume

we get,

Back substituting the value of t we get,

### Question:5 Integrate the functions

Given to integrate  function,

Let us assume

we get,

Now, by back substituting the value of t,

### Question:6 Integrate the functions

Given to integrate  function,

Let us assume

we get,

Now, by back substituting the value of t,

### Question:7 Integrate the functions

Given function ,

Assume the 19634

Back substituting the value of t in the above equation.

or,  , where C is any constant value.

### Question:8 Integrate the functions

Given function  ,

Assume the

Or

, where C is any constant value.

### Question:9 Integrate the functions

Given function  ,

Assume the

Now, back substituting the value of t in the above equation,

, where C is any constant value.

### Question:10 Integrate the functions

Given function  ,

Can be written in the form:

Assume the

, where C is any constant value.

### Question:11 Integrate the functions  , x > 0

Given function  ,

Assume the   so,

, where C is any constant value.

### Question:12 Integrate the functions

Given function  ,

Assume the

, where C is any constant value.

### Question:13  Integrate the functions

Given function  ,

Assume the

, where C is any constant value.

### Question:14  Integrate the functions

Given function  ,

Assume the

, where C is any constant value.

### Question:15  Integrate the functions

Given function  ,

Assume the

Now back substituting the value of t ;

, where C is any constant value.

### Question:16  Integrate the functions

Given function  ,

Assume the

Now back substituting the value of t ;

, where C is any constant value.

### Question:17  Integrate the functions

Given function  ,

Assume the

, where C is any constant value.

### Question:18  Integrate the functions

Given,

Let's do the following substitution

### Question:19  Integrate the functions

Given function  ,

Simplifying it by dividing both numerator and denominator by , we obtain

Assume the

Now, back substituting the value of t,

, where C is any constant value.

### Question:20 Integrate the functions

Given function  ,

Assume the

Now, back substituting the value of t,

, where C is any constant value.

### Question:21  Integrate the functions

Given function  ,

Assume the

Now, back substituting the value of t,

or  , where C is any constant value.

### Question:22 Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:23  Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:24 Integrate the functions

Given function  ,

or simplified as

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:25  Integrate the functions

Given function  ,

or simplified as

Assume the

Now, back substituted the value of t.

where C is any constant value.

### Question:26  Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:27  Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:28 Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:29  Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:30 Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:31  Integrate the functions

Given function  ,

Assume the

Now, back substituted the value of t.

, where C is any constant value.

### Question:32 Integrate the functions

Given function

Assume that

Now solving the assumed integral;

Now, to solve further we will assume

Or,

Now, back substituting the value of t,

### Question:33 Integrate the functions

Given function

Assume that

Now solving the assumed integral;

Now, to solve further we will assume

Or,

Now, back substituting the value of t,

### Question:34  Integrate the functions

Given function

Assume that

Now solving the assumed integral;

Multiplying numerator and denominator by ;

Now, to solve further we will assume

Or,

Now, back substituting the value of t,

### Question:35  Integrate the functions

Given function

Assume that

Now, back substituting the value of t,

### Question:36  Integrate the functions

Given function

Simplifying to solve easier;

Assume that

Now, back substituting the value of t,

### Question:37  Integrate the functions

Given function

Assume that

......................(1)

Now to solve further we take

So, from the equation (1), we will get

Now back substitute the value of u,

and then back substituting the value of t,

### Question:38  Choose the correct answer

Given integral

Taking the denominator

Now differentiating both sides we get

Back substituting the value of t,

Therefore the correct answer is D.

### Question:39  Choose the correct answer

Given integral

Therefore, the correct answer is B.

CBSE NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.3

### Question:1  Find the integrals of the functions

using the trigonometric identity

we can write the given question as

### Question:2  Find the integrals of the functions

Using identity

, therefore the given integral can be written as

### Question:3  Find the integrals of the functions

Using identity

Again use the same identity mentioned in the first line

### Question:4 Find the integrals of the functions

The integral can be written as

Let

Now,  replace the value of t, we get;

### Question:5 Find the integrals of the functions

rewrite the integral as follows

Let

......(replace the value of t as )

### Question:6 Find the integrals of the functions

Using the formula

we can write the integral as follows

### Question:7  Find the integrals of the functions

Using identity

we can write the following integral as

=

### Question:8  Find the integrals of the functions

We know the identities

Using the above relations we can write

### Question:9 Find the integrals of the functions

The integral is rewritten using trigonometric identities

### Question:10  Find the integrals of the functions

can be written as follows using trigonometric identities

Therefore,

### Question:11  Find the integrals of the functions

now using the identity

now using the below two identities

the value

.

the integral of the given function can be written as

### Question:12 Find the integrals of the functions

Using trigonometric identities we can write the given integral as follows.

### Question:13  Find the integrals of the functions

We know that,

Using this identity we can rewrite the given integral as

### Question:14  Find the integrals of the functions

Now,

### Question:15  Find the integrals of the functions

Therefore integration of  =
.....................(i)
Let assume

So, that
Now, the equation (i) becomes,

### Question:16  Find the integrals of the functions

the given question can be rearranged using trigonometric identities

Therefore, the integration of  = ...................(i)
Considering only
let

now the final solution is,

### Question:17  Find the integrals of the functions

now splitting the terms we can write

Therefore, the integration of

### Question:18  Find the integrals of the functions

The integral of the above equation is

Thus after evaluation, the value of integral is tanx+ c

### Question:19 Find the integrals of the functions

Let
We can write 1 =
Then, the equation can be written as

put the value of tan = t
So, that

### Question:20 Find the integrals of the functions

we know that
therefore,

let
Now the given integral can be written as

### Question:21  Find the integrals of the functions

using the trigonometric identities we can evaluate the following integral as follows

### Question:22  Find the integrals of the functions

Using the trigonometric identities following integrals can be simplified as follows

### Question:23 Choose the correct answer

The correct option is (A)

On reducing the above integral becomes

### Question:24 Choose the correct answer

The correct option is (B)

Let .
So,
(1+)

therefore,

NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.4

### Question:1 Integrate the functions

The given integral can be calculated as follows

Let
, therefore,

### Question:2  Integrate the functions

let suppose 2x = t
therefore 2dx = dt

.................using formula

### Question:3  Integrate the functions

let suppose 2-x =t
then, -dx =dt

using the identity

### Question:4  Integrate the functions

Let assume 5x =t,
then 5dx = dt

The above result is obtained using the identity

### Question:5  Integrate the functions

Let

The integration can be done as follows

### Question:6  Integrate the functions

let
then

using the special identities we can simplify the integral as follows

### Question:7  Integrate the functions

We can write above eq as
............................................(i)

for                    let

Now, by using eq (i)

### Question:8  Integrate the functions

The integration can be down as follows

let

........................using

### Question:9 Integrate the functions

The integral can be evaluated as follows

let

### Question:10 Integrate the functions

the above equation can be also written as,

let 1+x = t
then dx = dt
therefore,

### Question:12  Integrate the functions

the denominator can be also written as,

therefore

Let x+3 = t
then dx =dt

......................................using formula

### Question:13  Integrate the functions

(x-1)(x-2) can be also written as

=

therefore

let suppose

Now,

.............by using formula

### Question:14  Integrate the functions

We can write denominator as

therefore

let

### Question:15 Integrate the functions

(x-a)(x-b) can be written as

let

So,

### Question:16  Integrate the functions

let

By equating the coefficient of x and constant term on each side, we get
A = 1 and B=0

Let

### Question:17 Integrate the functions

let
By comparing the coefficients and constant term on both sides, we get;

A=1/2 and B=2
then

### Question:18  Integrate the functions

let

By comparing the coefficients and constants we get the value of A and B

A =  and B =

NOW,

...........................(i)

put
Thus

### Question:19 Integrate the functions

let

By comparing the coefficients and constants on both sides, we get
A =3  and B =34

....................................(i)

Considering

let

Now consider

here the denominator can be also written as
Dr =

Now put the values of  and  in eq (i)

### Question:20 Integrate the functions

let

By equating the coefficients and constant term on both sides  we get

A = -1/2 and B = 4

(x+2) = -1/2(4-2x)+4

....................(i)

Considering

let

now,

put the value of and

### Question:21 Integrate the functions

...........(i)

take

let

considering

putting the values  in equation (i)

### Question:22 Integrate the functions

Let

By comparing the coefficients and constant term, we get;

A = 1/2 and B =4

..............(i)

put

### Question:23 Integrate the functions

let

On comparing, we get

A =5/2 and B = -7

...........................................(i)

put

### Question:24  Choose the correct answer

The correct option is (B)

the denominator can be written as
now,

### Question:25 Choose the correct answer

The following integration can be done as

The correct option is (B)

NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.5

Question:1  Integrate the rational functions

Given function

Partial function of this function:

Now, equating the coefficients of x and constant term, we obtain

On solving, we get

### Question:2  Integrate the rational functions

Given function

The partial function of this function:

Now, equating the coefficients of x and constant term, we obtain

On solving, we get

### Question:3 Integrate the rational functions

Given function

Partial function of this function:

.(1)

Now, substituting  respectively in equation (1), we get

That implies

### Question:4  Integrate the rational functions

Given function

Partial function of this function:

.....(1)

Now, substituting  respectively in equation (1), we get

That implies

### Question:5  Integrate the rational functions

Given function

Partial function of this function:

...........(1)

Now, substituting  respectively in equation (1), we get

That implies

### Question:6  Integrate the rational functions

Given function

Integral is not a proper fraction so,

Therefore, on dividing  by , we get

Partial function of this function:

...........(1)

Now, substituting   respectively in equation (1), we get

No, substituting in equation (1) we get

### Question:7 Integrate the rational functions

Given function

Partial function of this function:

Now, equating the coefficients of  and the constant term, we get

and

On solving these equations, we get

From equation (1), we get

Now, consider

and we will assume

So,

or

### Question:8 Integrate the rational functions

Given function

Partial function of this function:

Now, putting  in the above equation, we get

By equating the coefficients of  and constant term, we get

then after solving, we get

Therefore,

### Question:9  Integrate the rational functions

Given function

can be rewritten as

Partial function of this function:

................(1)

Now, putting  in the above equation, we get

By equating the coefficients of  and  , we get

then after solving, we get

Therefore,

### Question:10 Integrate the rational functions

Given function

can be rewritten as

The partial function of this function:

Equating the coefficients of , we get

Therefore,

### Question:11 Integrate the rational functions

Given function

can be rewritten as

The partial function of this function:

Now, substituting the value of  respectively in the equation above, we get

Therefore,

### Question:12 Integrate the rational functions

Given function

As the given integral is not a proper fraction.

So, we divide  by , we get

can be rewritten as

....................(1)

Now, substituting  in equation (1), we get

Therefore,