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

Last updated date: 19th Sep 2024
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## NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1 - FREE PDF Download

NCERT Solutions for Ex 7.1 Class 12 starts with the introduction of Integral. This chapter covers the two types of Integrals i.e. Definite Integral and Indefinite Integral. The connection between these two types of integral is called the fundamental theorem of Calculus. The other topics under this chapter are integration as an inverse process of differentiation, geometrical interpretation of indefinite integral, properties of indefinite integral, and comparison between differentiation and integration. Below are some brief excerpts of the theory that will help you in solving the anti derivatives and integrals in Class 12 Ex 7.1. You can download the NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1 for FREE from Vedantu’s website.

Table of Content
1. NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1 - FREE PDF Download
2. Glance on NCERT Solutions Maths Chapter 7 Exercise 7.1 Class 12 | Vedantu
3. Formulas Used in Class 12 Maths 7.1
4. Access NCERT Solutions for Maths Class 12 Chapter 7 Integrals
5. Class 12 Maths Chapter 7: Exercises Breakdown
6. CBSE Class 12 Maths Chapter 7 Other Study Materials
7. Chapter-Specific NCERT Solutions for Class 12 Maths
FAQs

## Glance on NCERT Solutions Maths Chapter 7 Exercise 7.1 Class 12 | Vedantu

• In Class 12 Maths Ex 7.1, We will be solving for the anti derivatives (indefinite integrals) of various functions using the inspection method. This method involves recognizing the derivative of a standard function and then reversing the differentiation process to arrive at the original function.

• Integration as an Inverse Process of Differentiation

• There are 22 questions in Ex 7.1 class 12 maths which are fully solved by experts at Vedantu.

## Formulas Used in Class 12 Maths 7.1

• Standard Integrals: These are the basic building blocks of integration. Some commonly encountered standard integrals include:

• Integral of 1 (dx) = x

• Integral of x^n (dx) = x^(n+1)/(n+1), where n ≠ -1

Competitive Exams after 12th Science

## Access NCERT Solutions for Maths Class 12 Chapter 7 Integrals

### Exercise 7.1

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

sin $2x$

Ans:  We use the method of inspection as follows:

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

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

Thus, the anti-derivative of sin $2x$ is $-\frac{1}{2}\cos 2x$.

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

cos $3x$

Ans: We use the method of inspection as follows:

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

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

Thus, the anti - derivative of cos $3x$ is $\frac{1}{3}\sin 3x$.

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

${{e}^{2x}}$

Ans: We use the method of inspection as follows:

$\frac{d}{dx}\left( {{e}^{2x}} \right)\Rightarrow 2{{e}^{2x}}=\frac{1}{2}\frac{d}{dx}\left( {{e}^{2x}} \right)$

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

Thus, the anti-derivative of ${{e}^{2x}}$ is $\frac{1}{2}{{e}^{2x}}$.

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

${{\left( ax+b \right)}^{2}}$

Ans: We use the method of inspection as follows:

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

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

$\therefore {{\left( ax+b \right)}^{2}}=\frac{d}{dx}\left( \frac{1}{3a}{{\left( ax+b \right)}^{3}} \right)$

Thus, the anti-derivative of ${{\left( ax+b \right)}^{2}}$ is $\frac{1}{3a}{{\left( ax+b \right)}^{3}}$.

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

sin $2x-4{{e}^{3x}}$

Ans: We use the method of inspection as follows:

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

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

6. $\int{\left( 4{{e}^{3x}}+1 \right)dx}$

Ans:

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

$=4\int{{{e}^{3x}}dx}+\int{1}dx$

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

$=\frac{4}{3}{{e}^{3x}}+x+C$

7. $\int{{{x}^{2}}\left( 1-\frac{1}{{{x}^{2}}} \right)dx}$

Ans:

$\int{{{x}^{2}}\left( 1-\frac{1}{{{x}^{2}}} \right)dx}$

$=\int{\left( {{x}^{2}}-1 \right)dx}$

$=\frac{{{x}^{3}}}{3}-x+C$

8. $\int{\left( a{{x}^{2}}+bx+c \right)dx}$

Ans:

$\int{\left( a{{x}^{2}}+bx+c \right)}dx$

$=a\int{{{x}^{2}}dx+b\int{xdx+c\int{1.dx}}}$

$=a\left( \frac{{{x}^{3}}}{3} \right)+b\left( \frac{{{x}^{2}}}{2} \right)+cx+D$

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

9. $\int{\left( 2{{x}^{2}}+{{e}^{x}} \right)dx}$

Ans:

$\int{\left( 2{{x}^{2}}+{{e}^{x}} \right)dx}$

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

$=2\left( \frac{{{x}^{3}}}{3} \right)+{{e}^{x}}+C$

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

10. $\int{{{\left( \sqrt{x}-\frac{1}{\sqrt{x}} \right)}^{2}}dx}$

Ans:

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

$=\int{\left( x+\frac{1}{x}-2 \right)dx}$

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

$=\frac{{{x}^{2}}}{2}+\log \left| x \right|-2x+C$

11. $\int{\frac{{{x}^{3}}+5{{x}^{2}}-4}{{{x}^{2}}}dx}$

Ans: $\int{\frac{{{x}^{3}}+5{{x}^{2}}-4}{{{x}^{2}}}dx}$

$=\int{\left( x+5-4{{x}^{-2}} \right)dx}$

$=\int{xdx}+5\int{1.dx}-4\int{{{x}^{-2}}dx}$

$=\frac{{{x}^{2}}}{2}+5x+\frac{4}{x}+C$

12. $\int{\frac{{{x}^{3}}+3x+4}{\sqrt{x}}dx}$

Ans:$\int{\frac{{{x}^{3}}+3x+4}{\sqrt{x}}dx}$

$=\int{\left( {{x}^{\frac{5}{2}}}+3{{x}^{\frac{1}{2}}}+4{{x}^{-\frac{1}{2}}} \right)dx}$

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

$=\frac{2}{7}{{x}^{\frac{7}{2}}}+2{{x}^{\frac{3}{2}}}+8\sqrt{x}+C$

13. $\int{\frac{{{x}^{3}}-{{x}^{2}}+x-1}{x-1}dx}$

Ans: $\int{\frac{{{x}^{3}}-{{x}^{2}}+x-1}{x-1}dx}$

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

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

We obtain, on dividing:

$=\int{\left( {{x}^{2}}+1 \right)dx}$

$=\int{{{x}^{2}}dx}+\int{1.dx}$

$=\frac{{{x}^{3}}}{3}+x+C$

14. $\int{\left( 1-x \right)}\sqrt{x}dx$

Ans: $\int{\left( 1-x \right)}\sqrt{x}dx$

$=\int{\left( \sqrt{x}-{{x}^{\frac{3}{2}}} \right)dx}$

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

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

15. $\int{\sqrt{x}\left( 3{{x}^{2}}+2x+3 \right)dx}$

Ans: $\int{\sqrt{x}\left( 3{{x}^{2}}+2x+3 \right)dx}$

$=3\int{\left( {{x}^{\frac{5}{2}}}+2{{x}^{\frac{3}{2}}}+3{{x}^{\frac{1}{2}}} \right)}$  $=3\int{{{x}^{\frac{5}{2}}}dx+2\int{{{x}^{\frac{3}{2}}}dx+3\int{{{x}^{\frac{1}{2}}}}dx}}$  $=\frac{6}{7}{{x}^{\frac{7}{2}}}+\frac{4}{5}{{x}^{\frac{5}{2}}}+2{{x}^{\frac{3}{2}}}+C$

16. $\int{\left( 2x-3\cos x+{{e}^{x}} \right)dx}$

Ans: $\int{\left( 2x-3\cos x+{{e}^{x}} \right)dx}$

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

$=\frac{2{{x}^{2}}}{2}-3\left( \sin x \right)+{{e}^{x}}+C$

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

17. $\int{\left( 2{{x}^{2}}-3\sin x+5\sqrt{x} \right)}dx$

Ans: $\int{\left( 2{{x}^{2}}-3\sin x+5\sqrt{x} \right)}dx$

$=2\int{{{x}^{2}}dx-3\int{\sin xdx+5\int{{{x}^{\frac{1}{2}}}}dx}}$

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

$=\frac{2}{3}{{x}^{3}}+3\cos x+\frac{10}{3}{{x}^{\frac{3}{2}}}+C$

18. $\int{\sec x\left( \sec x+\tan x \right)dx}$

Ans: $\int{\sec x\left( \sec x+\tan x \right)dx}$

$=\int{\left( {{\sec }^{2}}x+\sec x\tan x \right)dx}$

$=\int{{{\sec }^{2}}xdx+\int{\sec x\tan xdx}}$

$=\tan x+\sec x+C$

19. $\int{\frac{{{\sec }^{2}}x}{\cos e{{c}^{2}}x}dx}$

Ans: $\int{\frac{{{\sec }^{2}}x}{\cos e{{c}^{2}}x}dx}$

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

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

$=\int{{{\tan }^{2}}xdx}$

$=\int{{{\sec }^{2}}xdx}-\int{1dx}$

$=\tan x-x+C$

20. $\int{\frac{2-3\sin x}{{{\cos }^{2}}x}dx}$

Ans: $\int{\frac{2-3\sin x}{{{\cos }^{2}}x}dx}$

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

$=\int{2{{\sec }^{2}}xdx}-3\int{\tan x\sec xdx}$

$=2\tan x-3\sec x+C$

21. The anti – derivative of $\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)$ equals

1. $\frac{1}{3}{{x}^{\frac{1}{3}}}+2{{x}^{\frac{1}{2}}}+C$

2. $\frac{2}{3}{{x}^{\frac{2}{3}}}+\frac{1}{2}{{x}^{2}}+C$

3. $\frac{2}{3}{{x}^{\frac{3}{2}}}+2{{x}^{\frac{1}{2}}}+C$

4. $\frac{3}{2}{{x}^{\frac{3}{2}}}+\frac{1}{2}{{x}^{\frac{1}{2}}}+C$

Ans:

$\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)$  $=\int{{{x}^{\frac{1}{2}}}dx}+\int{{{x}^{-\frac{1}{2}}}}dx=\frac{{{x}^{\frac{3}{2}}}}{\frac{3}{2}}+\frac{{{x}^{\frac{1}{2}}}}{\frac{1}{2}}+C$

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

Thus, the correct answer is C.

22. If $\frac{d}{dx}f\left( x \right)=4{{x}^{3}}-\frac{3}{{{x}^{4}}}$ such that $f\left( 2 \right)=0$ then $f\left( x \right)$ is

1. ${{x}^{4}}+\frac{1}{{{x}^{3}}}-\frac{129}{8}$

2. ${{x}^{3}}+\frac{1}{{{x}^{4}}}+\frac{129}{8}$

3. ${{x}^{4}}+\frac{1}{{{x}^{3}}}+\frac{129}{8}$

4. ${{x}^{3}}+\frac{1}{{{x}^{4}}}-\frac{129}{8}$

Ans: Given, $\frac{d}{dx}f\left( x \right)=4{{x}^{3}}-\frac{3}{{{x}^{4}}}$

Anti-derivative of $4{{x}^{3}}-\frac{3}{{{x}^{4}}}=f\left( x \right)$

$\therefore f\left( x \right)=\int{4{{x}^{3}}-\frac{3}{{{x}^{4}}}=f\left( x \right)}$

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

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

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

$Also,$

$f\left( 2 \right)=0$

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

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

$\Rightarrow C=\frac{-129}{8}$

$\therefore f\left( x \right)={{x}^{4}}+\frac{1}{{{x}^{3}}}-\frac{129}{8}$

Thus, the correct answer is A.

## Conclusion

In conclusion, Exercise 7.1 Class 12 Maths is a crucial stepping stone in mastering the concept of integrals, a fundamental aspect of calculus. By focusing on the basics of indefinite integrals and their standard formulas, Ex 7.1 Class 12 helps students build a strong foundation. Completing these problems not only enhances problem-solving skills but also prepares students for more advanced integration techniques in subsequent exercises and chapters.

Practice and ensure a thorough understanding, as the concepts learned here are integral to various applications in mathematics and beyond. Keep practicing, and students will find themselves well-prepared for more complex calculus challenges ahead!

## Class 12 Maths Chapter 7: Exercises Breakdown

 Exercise Number of Questions Exercise 7.2 39 Questions & Solutions (37 Short Answers, 2 MCQs) Exercise 7.3 24 Questions & Solutions (22 Short Answers, 2 MCQs) Exercise 7.4 25 Questions & Solutions (23 Short Answers, 2 MCQs) Exercise 7.5 23 Questions & Solutions (21 Short Answers, 2 MCQs) Exercise 7.6 24 Questions & Solutions (22 Short Answers, 2 MCQs) Exercise 7.7 11 Questions & Solutions (9 Short Answers, 2 MCQs) Exercise 7.8 22 Questions & Solutions (20 Short Answers, 2 MCQs) Exercise 7.9 10 Questions & Solutions (8 Short Answers, 2 MCQs) Exercise 7.10 21 Questions & Solutions (19 Short Answers, 2 MCQs) Miscellaneous Exercise 40 Questions & Solutions

## Chapter-Specific NCERT Solutions for Class 12 Maths

Given below are the chapter-wise NCERT Solutions for Class 12 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

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

1. Can you give an example of a standard formula used in exercise 7.1?

Yes, the standard formula used in Class 12 Maths Chapter 7 Exercise 7.1 solutions is the integral of a power function: ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.

2. What should I focus on while practicing Exercise 7.1?

Focus on mastering the basic formulas and understanding the properties of indefinite integrals, as this will help you solve more complex problems in later exercises.

3. What is the constant of integration, and why is it important?

The constant of integration, denoted by 'C', represents the family of all antiderivatives of a function. It is important because it accounts for the indefinite nature of integrals.