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NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7 - Integrals

NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7 - Integrals

Edited By Ramraj Saini | Updated on Dec 03, 2023 09:41 PM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.1

NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7 Integrals are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT solutions for Class 12 Maths chapter 7 exercise 7.1 is the first exercise of chapter 7 Integrals. Basic concepts of integrals are discussed in this exercise. This exercise includes concepts pertaining to finding out integrals of basic functions like sinx, cosx etc. Exercise 7.1 Class 12 Maths can be a good source to grasp the initial concepts of integrals. Head Start with basic concepts is a must in Integrals to reach an advanced level which can be easily learnt from NCERT Solutions for Class 12 Maths chapter 7 exercise 7.1 provided below. Also it is a good source to score well in CBSE Cass 12 Board Exam. Basic integration problems are also asked in competitive exams like JEE Main.

In subsequent exercises of Class 11 Maths NCERT book, students will cover advanced problems related to finding the area of a curve etc. 12th class Maths exercise 7.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

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Access NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1

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Integrals Class 12 Chapter 7 Exercise: 7.1

Question:1 Find an anti derivative (or integral) of the following functions by the method of inspection. \sin 2x

Answer:

GIven \sin 2x ;

So, the anti derivative of \sin 2x is a function of x whose derivative is \sin 2x .

\frac{d}{dx}\left ( \cos 2x \right ) = -2\sin 2x

\implies \sin 2x =\frac{-1}{2} \frac{d}{dx}\left ( \cos 2x \right )

Therefore, we have \implies \sin 2x = \frac{d}{dx}\left ( \frac{-1}{2}\cos 2x \right )

Or, antiderivative of \sin 2x is \left ( \frac{-1}{2}\cos 2x \right ) .

Question:2 Find an anti derivative (or integral) of the following functions by the method of inspection. \cos 3x

Answer:

GIven \cos 3x ;

So, the antiderivative of \cos 3x is a function of x whose derivative is \cos 3x .

\frac{d}{dx}\left ( \sin 3x \right ) = 3\cos3x

\implies \cos 3x =\frac{1}{3} \frac{d}{dx}\left ( \sin 3x \right )

Therefore, we have the anti derivative of \cos 3x is \frac{1}{3}\sin 3x .

Question:3 Find an anti derivative (or integral) of the following functions by the method of inspection. e ^{2x}

Answer:

GIven e ^{2x} ;

So, the anti derivative of e ^{2x} is a function of x whose derivative is e ^{2x} .

\frac{d}{dx}\left ( e ^{2x}\right ) = 2e ^{2x}

\implies e ^{2x} = \frac{1}{2}\frac{d}{dx}(e ^{2x})

\therefore e ^{2x} = \frac{d}{dx}(\frac{1}{2}e ^{2x})

Therefore, we have the anti derivative of e^{2x} is \frac{1}{2}e ^{2x} .

Question:4 Find an anti derivative (or integral) of the following functions by the method of inspection. ( ax + b )^2

Answer:

GIven ( ax + b )^2 ;

So, the anti derivative of ( ax + b )^2 is a function of x whose derivative is ( ax + b )^2 .

\frac{d}{dx} (ax+b)^3 = 3a(ax+b)^2

\Rightarrow (ax+b)^2 =\frac{1}{3a}\frac{d}{dx}(ax+b)^3

\therefore (ax+b)^2 = \frac{d}{dx}[\frac{1}{3a}(ax+b)^3]

Therefore, we have the anti derivative of (ax+b)^2 is [\frac{1}{3a}(ax+b)^3] .

Question:5 Find an anti derivative (or integral) of the following functions by the method of inspection. \sin 2x - 4 e ^{3x}

Answer:

GIven \sin 2x - 4 e ^{3x} ;

So, the anti derivative of

\frac{d}{dx} (-\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x}) = \sin 2x -4e^{3x}

Therefore, we have the anti derivative of \sin 2x - 4 e ^{3x} is \left ( -\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x} \right ) .

Question:6 Find the following integrals

\int ( 4e ^{3x}+1) dx

Answer:

Given intergral \int ( 4e ^{3x}+1) dx ;

4\int e ^{3x} dx + \int 1 dx = 4\left ( \frac{e^{3x}}{3} \right ) +x +C

or \frac{4}{3} e^{3x} +x +C , where C is any constant value.

Question:7 Find the following integrals \int x ^2 ( 1- \frac{1}{x^2})dx

Answer:

Given intergral \int x ^2 ( 1- \frac{1}{x^2})dx ;

\int x^2 dx - \int1dx

or \frac{x^3}{3} - x +C , where C is any constant value.

Question:8 Find the following integrals \int ( ax ^2 + bx + c ) dx

Answer:

Given intergral \int ( ax ^2 + bx + c ) dx ;

\int ax^2\ dx + \int bx\ dx + \int c\ dx

= a\int x^2\ dx + b\int x\ dx + c\int dx

= a\frac{x^3}{3}+b\frac{x^2}{2}+cx +C

or \frac{ax^3}{3}+\frac{bx^2}{2}+cx +C , where C is any constant value.

Question:9 Find the following integrals intergration of \int \left ( 2x^2 + e ^x \right ) dx

Answer:

Given intergral \int \left ( 2x^2 + e ^x \right ) dx ;

\int 2x^2\ dx + \int e^{x}\ dx

= 2\int x^2\ dx + \int e^{x}\ dx

= 2\frac{x^3}{3}+e^{x} +C

or \frac{2x^3}{3}+e^{x} +C , where C is any constant value.

Question:10 Find the following integrals \int \left ( \sqrt x - \frac{1}{\sqrt x } \right ) ^2 dx

Answer:

Given integral \int \left ( \sqrt x - \frac{1}{\sqrt x } \right ) ^2 dx ;

or \int (x+\frac{1}{x}-2)\ dx

= \int x\ dx + \int \frac{1}{x}\ dx -2\int dx

= \frac{x^2}{2} + \ln|x| -2x +C , where C is any constant value.

Question:11 Find the following integrals intergration of \int \frac{x^3 + 5x^2 - 4}{x^2} dx

Answer:

Given intergral \int \frac{x^3 + 5x^2 - 4}{x^2} dx ;

or \int \frac{x^3}{x^2}\ dx+\int \frac{5x^2}{x^2}\ dx -4\int \frac{1}{x^2}\ dx

\int x\ dx + 5\int1. dx - 4\int x^{-2}\ dx

= \frac{x^2}{2}+5x-4\left ( \frac{x^{-1}}{-1} \right )+C

Or, \frac{x^2}{2}+5x+\frac{4}{x}+C , where C is any constant value.

Question:12 Find the following integrals \int \frac{x^3+ 3x +4 }{\sqrt x } dx

Answer:

Given intergral \int \frac{x^3+ 3x +4 }{\sqrt x } dx ;

or \int \frac{x^3}{x^{\frac{1}{2}}}\ dx+\int \frac{3x}{x^{\frac{1}{2}}}\ dx +4\int \frac{1}{x^{\frac{1}{2}}}\ dx

= \int x^{\frac{5}{2}}\ dx + 3\int x^{\frac{1}{2}}\ dx +4\int x^{-\frac{1}{2}}\ dx

=\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+\frac{3\left ( x^{\frac{3}{2}} \right )}{\frac{3}{2}}+\frac{4\left ( x^{\frac{1}{2}} \right )}{\frac{1}{2}} +C

Or, = \frac{2}{7}x^{\frac{7}{2}} +2x^{\frac{3}{2}}+8\sqrt{x} +C , where C is any constant value.

Question:13 Find the following integrals intergration of \int \frac{x^3 - x^2 + x -1 }{x-1 } dx

Answer:

Given integral \int \frac{x^3 - x^2 + x -1 }{x-1 } dx

It can be written as

= \int \frac{x^2(x-1)+(x+1)}{(x-1)} dx

Taking (x-1) common out

= \int \frac{(x-1)(x^2+1)}{(x-1)} dx

Now, cancelling out the term (x-1) from both numerator and denominator.

= \int (x^2+1)dx

Splitting the terms inside the brackets

=\int x^2dx + \int 1dx

= \frac{x^3}{3}+x+c

Question:14 Find the following integrals \int (1-x) \sqrt x dx

Answer:

Given intergral \int (1-x) \sqrt x dx ;

\int \sqrt{x}\ dx - \int x\sqrt{x}\ dx or

\int x^{\frac{1}{2}}\ dx - \int x^{\frac{3}{2}} \ dx

= \frac{x^\frac{3}{2}}{\frac{3}{2}} - \frac{x^{\frac{5}{2}}}{\frac{5}{2}} +C

= \frac{2}{3}x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}+C , where C is any constant value.

Question:15 Find the following integrals \int \sqrt x ( 3x^2 + 2x +3 )dx

Answer:

Given intergral \int \sqrt x ( 3x^2 + 2x +3 )dx ;

= \int 3x^2\sqrt{x}\ dx + \int 2x\sqrt{x}\ dx + \int 3\sqrt {x}\ dx or = 3\int x^{\frac{5}{2}}\ dx + 2\int x^{\frac{3}{2}} \ dx +3\int x^{\frac{1}{2}} \ dx

= 3\frac{x^\frac{7}{2}}{\frac{7}{2}} +2\frac{x^{\frac{5}{2}}}{\frac{5}{2}} +3\frac{x^{\frac{3}{2}}}{\frac{3}{2}} +C

= \frac{6}{7}x^{\frac{7}{2}} + \frac{4}{5}x^{\frac{5}{2}}+ 2x^{\frac{3}{2}}+C , where C is any constant value.


Question:16 Find the following integrals \int ( 2x - 3 \cos x + e ^x ) dx

Answer:

Given integral \int ( 2x - 3 \cos x + e ^x ) dx ;

splitting the integral as the sum of three integrals

\int 2x\ dx -3 \int \cos x\ dx +\int e^{x}\ dx

= 2 \frac{x^2}{2} - 3 \sin x + e^x+C

= x^2 - 3 \sin x + e^x+C , where C is any constant value.

Question:17 Find the following integrals \int ( 2 x ^2 - 3 \sin x + 5 \sqrt x ) dx

Answer:

Given integral \int ( 2 x ^2 - 3 \sin x + 5 \sqrt x ) dx ;

2\int x^2\ dx -3\int \sin x\ dx + 5\int \sqrt {x}\ dx

= 2 \frac{x^3}{3} - 3(-\cos x ) +5\left ( \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ) +C

= \frac{2x^3}{3} +3\cos x +\frac{10}{3} x^{\frac{3}{2}}+C , where C is any constant value.

Question:18 Find the following integrals \int \sec x ( \sec x + \tan x ) dx

Answer:

Given integral \int \sec x ( \sec x + \tan x ) dx ;

\int (\sec^2x+ \sec x \tan x ) \ dx

Using the integral of trigonometric functions

= \int (sec^2 x )\ dx+ \int \sec x \tan x\ dx

= \tan x + \sec x +C , where C is any constant value.

Question:19 Find the following integrals intergration of \int \frac{sec ^2 x }{cosec ^2 x } dx

Answer:

Given integral \int \frac{sec ^2 x }{cosec ^2 x } dx ;

\int \frac{\frac{1}{\cos^2x}}{\frac{1}{\sin^2 x}}\ dx

= \int \frac{\sin^2 x }{\cos ^2 x } \ dx

=\int (\sec^2 x-1 )\ dx

=\int \sec^2 x\ dx-\int1 \ dx

= \tan x -x+C , where C is any constant value.

Question:20 Find the following integrals \int \frac{2- 3 \sin x }{\cos ^ 2 x } dx

Answer:

Given integral \int \frac{2- 3 \sin x }{\cos ^ 2 x } dx ;

\int \left ( \frac{2}{\cos^2x}-\frac{3\sin x }{\cos^2 x} \right )\ dx

Using antiderivative of trigonometric functions

= 2\tan x -3\sec x +C , where C is any constant value.

Question:21 Choose the correct answer
The anti derivative of \left ( \sqrt x + 1/ \sqrt x \right ) equals

A) \frac{1}{3}x ^{1/3} + 2 x ^{1/2}+ C \\\\ B) \frac{2}{3}x ^{2/3} + \frac{1}{2}x ^{2}+ C \\\\ C ) \frac{2}{3}x ^{3/2} + 2 x ^{1/2}+ C\\\\ D) \frac{3}{2}x ^{3/2} + \frac{1}{2} x ^{1/2}+ C

Answer:

Given to find the anti derivative or integral of \left ( \sqrt x + 1/ \sqrt x \right ) ;

\int \left ( \sqrt x + 1/ \sqrt x \right )\ dx

\int x^{\frac{1}{2}}\ dx + \int x^{-\frac{1}{2}}\ dx

= \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+C

= \frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} +C , where C is any constant value.

Hence the correct option is (C).

Question:22 Choose the correct answer The anti derivative of

If \frac{d}{dx}f(x) = 4 x ^3 - \frac{3}{x^4} such that f (2) = 0. Then f (x) is

A ) x ^ 4 + \frac{1}{x^3} - \frac{129 }{8} \\\\ B ) x ^ 3 + \frac{1}{x^4} - \frac{129 }{8} \\\\ C ) x ^ 4 + \frac{1}{x^3} + \frac{129 }{8}\\\\ D) x ^ 3 + \frac{1}{x^4} - \frac{129 }{8}

Answer:

Given that the anti derivative of \frac{d}{dx}f(x) = 4 x ^3 - \frac{3}{x^4}

So, \frac{d}{dx}f(x) = 4 x ^3 - \frac{3}{x^4}

f(x) = \int 4 x ^3 - \frac{3}{x^4}\ dx

f(x) = 4\int x ^3 - 3\int {x^{-4}}\ dx

f(x) = 4\left ( \frac{x^4}{4} \right ) -3\left ( \frac{x^{-3}}{-3} \right )+C

f(x) = x^4+\frac{1}{x^3} +C

Now, to find the constant C;

we will put the condition given, f (2) = 0

f(2) = 2^4+\frac{1}{2^3} +C = 0

16+\frac{1}{8} +C = 0

or C = \frac{-129}{8}

\Rightarrow f(x) = x^4+\frac{1}{x^3}-\frac{129}{8}

Therefore the correct answer is A .

More About NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1

The NCERT Class 12 Maths chapter Integrals covers a total of 12 exercises including one miscellaneous exercise. Exercise 7.1 Class 12 Maths provides solutions to 22 main questions and their sub-questions. It includes basic questions related to finding integrals of basic functions. NCERT Solutions for Class 12 Maths chapter 7 exercise 7.1 must be referred to get a strong command on integrals.

Also Read| Integrals Class 12 Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1

  • The NCERT syllabus Class 12th Maths chapter 7 exercise provided here is in detail which is solved by subject matter experts .
  • Practicing exercise 7.1 Class 12 Maths can help students in a tremendous way to prepare for exams.
  • These Class 12 Maths chapter 7 exercise 7.1 solutions can be asked directly in the Board exams.
  • NCERT solutions for Class 12 Maths chapter 7 exercise 7.1 are highly recommended to students and can be used to solve physics questions also for the related concepts.
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Key Features Of NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7

  • Comprehensive Coverage: The solutions encompass all the topics covered in ex 7.1 class 12, ensuring a thorough understanding of the concepts.
  • Step-by-Step Solutions: In this class 12 maths ex 7.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
  • Accuracy and Clarity: Solutions for class 12 ex 7.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
  • Conceptual Clarity: In this 12th class maths exercise 7.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
  • Inclusive Approach: Solutions for ex 7.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
  • Relevance to Curriculum: The solutions for class 12 maths ex 7.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.
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Frequently Asked Questions (FAQs)

1. What is Integration ?

Integration is the process of finding the antiderivative of a function which gives the area of the curve formed by a function. The integration is the inverse process of differentiation.

2. Where Integration is used ?

Integration is used to find the area, volume etc. of many functions. 

3. What is the importance of the Integrals chapter in Board exams ?

The Integral chapter covers a large portion of Class 12 Maths syllabus. It has applications in Physics as well. Hence it is advisable to students to not leave this chapter at any cost. 

4. What is the difficulty level of questions of this chapter?

In board, easy to moderate level of questions are asked but in competitive exams, advanced level of problems are asked which can only be tackled with practice. 

5. Mention some topics in Exercise 7.1 Class 12 Maths.

It includes basic topics related to finding the integration of basic functions. Eg sinx, cosx etc. 

6. Mention the total number of questions in this exercise ?

There are 22 questions in this exercise. For more questions students can refer to NCERT exemplar problems

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Have a question related to CBSE Class 12th ?
i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

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Ashvadha 12 July,2024
Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

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kumardurgesh1802 10 July,2024
I have completed my 12th with arts cbse. I want to repeat my 12th with pcmb. How to become a private candidate and what are deadline to be able to appear in 2025 exam?

Hello student,

If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:

  • No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
  • You have to appear for the 2025 12th board exams.
  • Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
  • Aim to register before late October to avoid extra fees.
  • Schools might not offer classes for private students, so focus on self-study or coaching.

Remember , these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

Good luck with your studies!

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kumardurgesh1802 10 July,2024
I passed my 12th cbse boards in 2023 , and first time appear for jee mains and advance in 2024 so , am i eligible for 2025 jee mains and advance ?

Hello Aspirant , Hope your doing great . As per your query , your eligible for JEE mains in the year of 2025 , Every candidate can appear for the JEE Main exam 6 times over three consecutive years . The JEE Main exam is held two times every year, in January and April.

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satyamercedes0626 23 July,2024
I've passed my class 12th board exam from cbse in PCB in 2024. Now I'll give mathematics exam from cbse as a additional subject in 2025. However, they'll provide separate marksheet for maths. Will i be eligible for jee?? And how I'll fill the form??

Hi there,

Hope you are doing fine

Yes you are certainly eligible for giving the jee exam in the year 2025. You must pass the maths exam with at least 75% criteria as required by jee and provide the marksheet and the passing certificate while registering for the exam.


Pursuing maths as an additional subject while taking biology as your main subject does not offer any hindrance in you appearing for the jee examination. It is indeed an privilege to pursue both maths and biology as the subjects and prepare for the same.

There will be no issue in filling the form while registering for the exam as it will only require your basic details and marksheet which you can provide by attaching the marksheet of maths also. Also, a detailed roadmap is also available on the official websites on how to fill the registration form. So you can fill the form easily.


Hope this resolves your query.

Read Complete Answer
Abhishek 12 July,2024

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