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# Class 12 maths NCERT Solutions Exercise 7.1-Integral

Class 12 maths NCERT Solutions Exercise 7.1 of Chapter 7-Integrals-Here NCERT solutions of class 12 maths exercise 7.1 of chapter 12 are the solutions of the chapter 7 exercise 7.1-integrals of class maths NCERT text book . All these NCERT solutions are well explained by the maths expert through a step by step method so that every student could be assessable to the exercise 7.1 of chapter 7-Integral. The purpose of these NCERT Solutions of exercise 7.1 is to clear the concept of integration among the students of class 12.

## Class 12 maths NCERT Solutions Exercise 7.1-Integral

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What is Integrals? Integration is just opposite to differentiation,to understand integration let’s have a function (3x +2), its derivative is 3.Then we can say integral of 3 is 3x +C where C =2 is known as an arbitrary constant. It can be cleared by the following examples.

y = 3x, dy/dx=3

y =3x +2,dy/dx =3

y= 3x +3,dy/dx =3

The differentiation of y  with respect to x is the same 3, so we can say that integration of 3 is  3x +C, where C may be any number, further, we can say that 3x +C is the indefinite integration(integral) of 3 , in other words, it can also be called that 3x+C is the anti-derivative of 3.

If there is a function f(x) such that its derivative is g(x),then f(x) is anti-derivative(Integral) of the function g(x).Therefore when the differentiation of a function is integrated we get the original function,in simple words integration is opposite of diffrentiation.

The d/dx(xn) = nxn-1

Here, integral of nxn-1 is xn

Integration is represented by the sign and dx is attached at the end of the function which means integration of the function with respect to x. The integral of a function is calculated as follows.

Application of integration: When differentiated parts of a region within limits combine together we get the integral of the function within the region. So, integration is used to calculate physical quantities like area, volume, mass, etc.It is widely used in science and technology for the evaluation of the different type of physical quantities and in business predicting the outcomes of particular inputs applied.

## Class 12 Maths NCERT solutions of Chapter 7 Integrals

Exercise 7.1- Integrals

Exercise 7.2-Integral

Exercise 7.3-Integrals

Exercise 7.4 -Integral

## NCERT Solutions Exercise 7.1-Integral

Q1.Find an anti-derivative (or integral) of the following functions by the method of inspection

Sin 2x

Ans.

We know -sinθ is the derivative of cosθ,so differentiation of cosθ is =-sin

Hence anti-derivative of sin2x is -1/2(cos2x).

Q2.Find an anti-derivative (or integral) of the following functions by the method of inspection

cos 3x

Ans. We know cosθ is the derivative of sinθ

Therefore anti-derivative (integral) of cos3x is 1/3(sin3x).

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Q3.Find an anti-derivative (or integral) of the following functions by the method of inspection

Ans.We know the derivative of is ,so

Therefore anti-derivative of is 1/2().

Q4.Find an anti-derivative (or integral) of the following functions by the method of inspection

(ax +b)²

Ans.We know the derivative of x³ is 3x², so

Therefore antiderivative of (ax+b)² is 1/3a(ax+b)³

### Class 12 Maths NCERT solutions of Chapter 7 Integrals

Q5.Find an anti-derivative (or integral) of the following functions by the method of inspection

Ans.We know -sinθ is the derivative of cosθ

We know derivative of  is

Subtracting equation (ii) from (i), we get

Therefore antiderivative of   is

Find the following integrals from Exercises 6  to 20.

Q6.

Ans.

Ans.

Ans.

### Class 12 Maths NCERT solutions of Chapter 7 Integrals

Ans.

Ans.

Writting the function in expanded form

Ans.

Ans.

### Class 12 Maths NCERT solutions of Chapter 7 Integrals

Ans.

Dividing  numerator by the denominator, we get x² +1

Ans.

+ C

Ans.

+ C

Ans.

Ans.

Ans. On multiplication, we get,

By taking separately, we get,

We get,

= tan x + sec x + C

Ans. We get,

So,

We get,

On further calculation, we get,

By taking seriously, we get

= tan x – x + C

### Class 12 Maths NCERT solutions of Chapter 7 Integrals

Ans.

Ans.

Therefore the correct answer is (C)

Ans. We are given

Therefore anti-derivative of     is f(x)

We are given f(2) =0

Substituting this value of C in f(x)

Therefore the correct answer is (A)

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## NCERT Solutions of Science and Maths for Class 9,10,11 and 12

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### NCERT Solutions for class 11 maths

 Chapter 1-Sets Chapter 9-Sequences and Series Chapter 2- Relations and functions Chapter 10- Straight Lines Chapter 3- Trigonometry Chapter 11-Conic Sections Chapter 4-Principle of mathematical induction Chapter 12-Introduction to three Dimensional Geometry Chapter 5-Complex numbers Chapter 13- Limits and Derivatives Chapter 6- Linear Inequalities Chapter 14-Mathematical Reasoning Chapter 7- Permutations and Combinations Chapter 15- Statistics Chapter 8- Binomial Theorem Chapter 16- Probability

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### NCERT solutions for class 12 maths

 Chapter 1-Relations and Functions Chapter 9-Differential Equations Chapter 2-Inverse Trigonometric Functions Chapter 10-Vector Algebra Chapter 3-Matrices Chapter 11 – Three Dimensional Geometry Chapter 4-Determinants Chapter 12-Linear Programming Chapter 5- Continuity and Differentiability Chapter 13-Probability Chapter 6- Application of Derivation CBSE Class 12- Question paper of maths 2021 with solutions Chapter 7- Integrals Chapter 8-Application of Integrals

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