NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Chapter 7 has been created by Kopykitab’s subject expertise to assist students in their 12th board exam preparations. The concept of integrals is covered in Chapter 7 of the NCERT Maths Book for Class 12. You will learn about integral calculus (definite and indefinite), its properties, and much more in this chapter. 

• All you need to know about NCERT Solutions Class 12 Maths

We are here with Integrals Class 12 NCERT Solutions for your 12th final exam. Download and go through the NCERT Solutions for Class 12 Maths Chapter 7 PDF!

Download NCERT Solutions for Class 12 Maths Chapter 7 Integrals

This chapter explains integral concepts in-depth and in an easy-to-understand manner. These integrals Class 12 solutions are very basic and can assist students in quickly grasping the problem-solving process. You can access these Integrals Class 12 NCERT Solutions from the below link and use them for free to practice on your own time.

Access Integrals Class 12 NCERT Solutions

NCERT-Solutions-Class-12-Maths-Chapter-7-Integrals pdf

 

Our subject specialists created these Chapter 7 NCERT Solutions for Class 12 Maths with the student’s needs in mind for the CBSE Board Examination. Get the class 12 maths chapter 7 NCERT solutions pdf on your device and start with your exam preparation. 

What will you get in NCERT Solutions for Class 12 Maths Chapter 7?

The chapter integrals are part of the Calculus unit, which accounts for 35 of the total points. This chapter contains 11 exercises as well as a miscellaneous exercise to help students understand the principle of Integrals. The following topics are covered in Indefinite Integrals Class 12 NCERT Solutions:

Subtopics to read in the chapter are –

7.1 Introduction

7.2 Integration as an Inverse Process of Differentiation

7.2.1 Geometrical interpretation of indefinite integral

7.2.2 Indefinite Integrals Class 12 NCERT Solutions: 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

7.8.1 Area 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

Exercise to practice in this chapter are as follows-

Exercise 7.1 Solutions 22 Questions

Exercise 7.2 Solutions 39 Questions

Exercise 7.3 Solutions 24 Questions

Exercise 7.4 Solutions 25 Questions

Exercise 7.5 Solutions 23 Questions

Exercise 7.6 Solutions 24 Questions

Exercise 7.7 Solutions 11 Questions

Exercise 7.8 Solutions 6 Questions

Exercise 7.9 Solutions 22 Questions

Exercise 7.10 Solutions 10 Questions

Exercise 7.11 Solutions 21 Questions

Miscellaneous Exercise On Chapter 7 Solutions 44 Questions

Let’s have brief look at the sections the chapter covers-

NCERT solutions for class 12 maths Chapter 7.1

Like every other chapter, this also starts with a sweet introduction that relaxes the minds of the student for a fresh start. The chapter begins with the historical backdrop of differential calculus and the introduction of integral calculus in the era. Students will get a vivid idea about derivatives and their role in defining tangent lines to the graph of the function. This topic will also state the methods of calculating the slopes of tangent lines.

With this chapter, students will get first-hand exposure to integral calculus, integration and differentiation,  and more in this regard.

Key highlights that this chapter covers are as follow-

• What are indefinite integrals and how they are denoted?
• What are definite integrals?
• How the relationship between the fundamentals of the indefinite integrals and definite integrals act as a practical tool for both science and engineering platforms. Well, this relationship is coined as the fundamental theorem of calculus.

Well, there is a lot to work your hand on the chapter, let’s get to it-

NCERT solutions for class 12 maths Chapter 7.2

This chapter teaches students the methods to evaluate integrals of a function with derivatives provided. And the other name of integrals is ‘anti-differentiation’. Students will come across terms such as ‘arbitrary constant’ which is ‘constant for integration’ and for any arbitrary real number. It is indefinite integrals. Also, you will get exposure to learning many new important formulas and properties in this segment thus offering you several new exciting topics.

NCERT solutions for class 12 maths Chapter 7.3

 This section lets students enjoy different methods of finding integral except the method of inspection, which we discussed earlier. Students will get hands-on experience in evaluating that inspection is not suitable for several kinds of functions. This puts in front three different ways of integration-

• By substitution
• Using partial function
• By parts

With time you will slowly get to delve deep into the topic where you will have to use many trigonometric identities such as-

sin²θ+cos²θ=1

2 sin θ cos θ= 2 sin θ

2 sin θ1 .cosθ 2= sin (θ 1+θ2)+ sin (θ1-θ2)

NCERT solutions for class 12 maths chapter 7.4

The section is equipped with many varieties of integral formulas and their application to many other standard integrals associated with them. We are here with a few shortcut methods which will make the sums of child’s play. Make sure to grasp the formulas listed in this section to have an easy understanding of the sums to follow.

NCERT solutions for class 12 maths chapter 7.5

This area expands the ideology that students learned in the 7.3 section thus disclosing more from integrals by partial functions. Students will also get to study improper rational functions, proper rational functions, disintegration, and more here.

NCERT solutions for class 12 maths chapter 7.6

Now, in this portion students will get in-depth knowledge about partial integration along with how to integrate product on functions,  and more to get the concept in elaboration. Students would also learn to apply integration on both the sides with a valid conclusion such as-

(1st function)*(integral of 2nd function)-Integral of [(differential coefficient of 1st function )* (integral of the second function)]

NCERT solutions for class 12 maths chapter 7.7

Students will have close exposure to the topic of definite integral in this section. Students will study the concept along with notations-  where ‘a’ is the lower limit and ‘b’ is the upper limit. Students will also learn about definite integrals as a limit of a sum or as an antiderivative in the intervals.

NCERT solutions for class 12 maths chapter 7.8

Students after coming to this section will get a subtle understanding that both integration and differentiation are interlinked notions. Well, this simple proposition derives antiderivative differences at the upper and lower limit of the integration process. Students will also study about-

• Area function- ∫abf(x)dx
• First fundamental theorem of integral calculus- A’(x)=f(x)
• Then second fundamental theorem of integral calculus

Make sure to read all the formulas to get confident with the same by exam.

NCERT solutions for class 12 maths chapter 7.9

 This section of the chapter teaches students about the specific steps taken to calculate the definite integrals ∫ab f(x)dx

 By substituting exactly the way mentioned below-

• Decrease the value of the given integral to familiar form by substituting y= f(x) and x= q(y) and consider the integral without limits.
• Now go for integrating the new integrand without mentioning the constant of integration.

NCERT solutions for class 12 maths chapter 7.10

This section covers the several properties connected with definite integrals thus simplifying the process of calculating definite integrals. Let’s have look at the properties given-

Properties	Description
Property 1	p∫q f(a) da = p∫q f(t) dt
Property 2	p∫q f(a) d(a) = – q∫p f(a) d(a), Also p∫p f(a) d(a) = 0
Property 3	p∫q f(a) d(a) = p∫r f(a) d(a) + r∫q f(a) d(a)
Property 4	p∫q f(a) d(a) = p∫q f( p + q – a) d(a)
Property 5	o∫p f(a) d(a) = o∫p f(p – a) d(a)
Property 6	∫02p f(a)da = ∫0p f(a)da +∫0p f(2p-a)da…if f(2p-a) = f(a)
Property 7	2 parts ●        ∫02 f(a)da = 2∫0a f(a) da … if f(2p-a) = f(a) ●        ∫02 p f(a)da = 0 … if f(2p-a) = -f(a)
Property 8	2 parts ●        ∫-pp f(a)da = 2∫0p f(a) da … if f(-a) = f(a) or it’s an even function ●        ∫-ppf(a)da = 0 … if f(2p-a) = -f(a) or it’s an odd function

NCERT Solutions for Class 12 Maths Chapter 7: Theorems and formulas to follow-

Let us have a sneak peek at the theorems and formulas this chapter gets loaded with-

Also Access:

Benefits NCERT Solutions for Class 12 Maths Chapter 7

We are here with the benefits of following NCERT Solutions for Class 12 Maths Chapter 7:

1. The easy language format of the NCERT solutions is a definite path that lets you finish the syllabus on time.
2. It will aid in the development of student confidence.
3. The content is well-structured to make learning and understanding easier for students.
4. This chapter will assist students in solidifying their understanding of integration.
5. Students may use NCERT solutions to help them with their assignments and competitive tests.

This is all about NCERT Solutions for Class 12 Maths Chapter 7 for you. To understand the chapter in a meticulous manner make sure to study NCERT Solutions for Class 12 Maths Chapter 7 for your upcoming CBSE. If you still come across any issue relevant to the same make sure to comment down to get our experts to answer your queries.

FAQs: Integrals Class 12 NCERT Solutions