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) = nx^n-1

Here, integral of nx^n-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

NCERT Solutions for class 9 maths

Chapter 1- Number System	Chapter 9-Areas of parallelogram and triangles
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3- Coordinate Geometry	Chapter 11-Construction
Chapter 4- Linear equations in two variables	Chapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s Geometry	Chapter 13-Surface Areas and Volumes
Chapter 6-Lines and Angles	Chapter 14-Statistics
Chapter 7-Triangles	Chapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science 

Chapter 1-Matter in our surroundings	Chapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?	Chapter 10- Gravitation
Chapter3- Atoms and Molecules	Chapter 11- Work and Energy
Chapter 4-Structure of the Atom	Chapter 12- Sound
Chapter 5-Fundamental unit of life	Chapter 13-Why do we fall ill ?
Chapter 6- Tissues	Chapter 14- Natural Resources
Chapter 7- Diversity in living organism	Chapter 15-Improvement in food resources
Chapter 8- Motion	Last years question papers & sample papers

NCERT Solutions for class 10 maths

Chapter 1-Real number	Chapter 9-Some application of Trigonometry
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3-Linear equations	Chapter 11- Construction
Chapter 4- Quadratic equations	Chapter 12-Area related to circle
Chapter 5-Arithmetic Progression	Chapter 13-Surface areas and Volume
Chapter 6-Triangle	Chapter 14-Statistics
Chapter 7- Co-ordinate geometry	Chapter 15-Probability
Chapter 8-Trigonometry

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Chapter 1- Chemical reactions and equations	Chapter 9- Heredity and Evolution
Chapter 2- Acid, Base and Salt	Chapter 10- Light reflection and refraction
Chapter 3- Metals and Non-Metals	Chapter 11- Human eye and colorful world
Chapter 4- Carbon and its Compounds	Chapter 12- Electricity
Chapter 5-Periodic classification of elements	Chapter 13-Magnetic effect of electric current
Chapter 6- Life Process	Chapter 14-Sources of Energy
Chapter 7-Control and Coordination	Chapter 15-Environment
Chapter 8- How do organisms reproduce?	Chapter 16-Management of Natural Resources

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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