Class 12 maths NCERT Solutions Exercise 7.1 of Chapter 7-Integrals - Future Study Point

Class 12 maths NCERT Solutions Exercise 7.1 of Chapter 7-Integrals

ex.7.1-Integrals

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.

ex.7.1-Integrals

 

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

NCERT Solutions for class 9 maths

Chapter 1- Number SystemChapter 9-Areas of parallelogram and triangles
Chapter 2-PolynomialChapter 10-Circles
Chapter 3- Coordinate GeometryChapter 11-Construction
Chapter 4- Linear equations in two variablesChapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s GeometryChapter 13-Surface Areas and Volumes
Chapter 6-Lines and AnglesChapter 14-Statistics
Chapter 7-TrianglesChapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science 

Chapter 1-Matter in our surroundingsChapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?Chapter 10- Gravitation
Chapter3- Atoms and MoleculesChapter 11- Work and Energy
Chapter 4-Structure of the AtomChapter 12- Sound
Chapter 5-Fundamental unit of lifeChapter 13-Why do we fall ill ?
Chapter 6- TissuesChapter 14- Natural Resources
Chapter 7- Diversity in living organismChapter 15-Improvement in food resources
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NCERT Solutions for class 10 maths

Chapter 1-Real numberChapter 9-Some application of Trigonometry
Chapter 2-PolynomialChapter 10-Circles
Chapter 3-Linear equationsChapter 11- Construction
Chapter 4- Quadratic equationsChapter 12-Area related to circle
Chapter 5-Arithmetic ProgressionChapter 13-Surface areas and Volume
Chapter 6-TriangleChapter 14-Statistics
Chapter 7- Co-ordinate geometryChapter 15-Probability
Chapter 8-Trigonometry

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

NCERT Solutions for class 11 maths

Chapter 1-SetsChapter 9-Sequences and Series
Chapter 2- Relations and functionsChapter 10- Straight Lines
Chapter 3- TrigonometryChapter 11-Conic Sections
Chapter 4-Principle of mathematical inductionChapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbersChapter 13- Limits and Derivatives
Chapter 6- Linear InequalitiesChapter 14-Mathematical Reasoning
Chapter 7- Permutations and CombinationsChapter 15- Statistics
Chapter 8- Binomial Theorem Chapter 16- Probability

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NCERT solutions for class 12 maths

Chapter 1-Relations and FunctionsChapter 9-Differential Equations
Chapter 2-Inverse Trigonometric FunctionsChapter 10-Vector Algebra
Chapter 3-MatricesChapter 11 – Three Dimensional Geometry
Chapter 4-DeterminantsChapter 12-Linear Programming
Chapter 5- Continuity and DifferentiabilityChapter 13-Probability
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Chapter 7- Integrals
Chapter 8-Application of Integrals

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