# NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.9) Exercise 7.9

The study of mathematics is a dynamic and complex subject. In its composition, it includes a wide variety of mathematical themes and concepts. It is important to note that each of these concepts and themes has its own unique underlying principles and working formulas. Furthermore, these concepts include a variety of theorems, subtopics, and specific methods and procedures of calculation that must be thoroughly analysed and understood. Numbers and numerical symbols are logically sequenced to reach certain results or to verify certain results. To be able to handle different types of calculations competently, students need to become familiar with the logical organization of steps. The importance of the practical application of knowledge implies this process of familiarization. Additionally, students should be informed about the many ways in which formulas can be applied in practice to solve problems. In the field of mathematics, calculations are not composed of random stacks of numbers and symbols.

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## NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.9) Exercise 7.9

Students are strongly advised to follow the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, for a number of reasons. Firstly, the solutions are provided by qualified people who strictly follow the CBSE pattern and syllabus. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 are provided and reviewed multiple times to make sure that the solutions are published correctly and that there are no mistakes, because having errors in the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 is entirely counterproductive.

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

The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, primarily cater to one of the most complicated topics of the entire syllabus: Integrals. Extramarks has allowed students to access these solutions, which can be accessed through the website. Like the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 that Extramarks has provided for the students, they have also released solutions for all other exercises in all other subjects as well. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, have been of great help to every student who has used them.

### Important Questions for Class 12 Maths Chapter 7

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### NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.9

Following the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, it is well understood that the exercise is completely based on the evaluation of definite integrals by substitution. This is almost the end of the integrals chapter, and therefore, this exercise abounds with problems which have references to the previous exercises; therefore, the explanations provided for these problems are very well explained, and students who have had a shaky introduction to these concepts have reported finding the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 extremely helpful.

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Q.1

$\begin{array}{l}\mathrm{Choose}\text{}\mathrm{the}\text{}\mathrm{correct}\text{}\mathrm{answer}\\ {\int }_{1}^{\sqrt{3}}\frac{\mathrm{dx}}{1+{\mathrm{x}}^{2}}\text{equals}\\ \left(\text{A}\right)\text{}\frac{\mathrm{\pi }}{3}\\ \left(\text{B}\right)\text{}\frac{2\mathrm{\pi }}{3}\\ \left(\text{C}\right)\text{}\frac{\mathrm{\pi }}{6}\\ \left(\text{D}\right)\text{}\frac{\mathrm{\pi }}{12}\end{array}$

Ans

$\begin{array}{l}\text{Let\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}={\int }_{1}^{\sqrt{3}}\frac{\mathrm{dx}}{1+{\mathrm{x}}^{2}}\\ \int \frac{1}{1+{\mathrm{x}}^{2}}\text{\hspace{0.17em}}\mathrm{dx}={\mathrm{tan}}^{-1}\mathrm{x}\\ \text{Therefore, by the second fundamental theorem, we have}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I}=\mathrm{F}\left(\sqrt{3}\right)-\mathrm{F}\left(1\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={\text{tan}}^{-1}\left(\sqrt{3}\right)-{\text{tan}}^{-1}\left(1\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{\mathrm{\pi }}{3}-\frac{\mathrm{\pi }}{4}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{4\mathrm{\pi }-3\mathrm{\pi }}{12}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{\mathrm{\pi }}{12}\\ \text{Hence, the correct option is D.}\end{array}$

Q.2

$\begin{array}{l}\mathrm{Choose}\mathrm{the}\mathrm{correct}\mathrm{answer}\\ {\int }_{0}^{\frac{2}{3}}\frac{\mathrm{dx}}{4+9{\mathrm{x}}^{2}}\text{equals}\\ \left(\text{A}\right)\frac{\mathrm{\pi }}{6}\\ \left(\text{B}\right)\frac{\mathrm{\pi }}{12}\\ \left(\text{C}\right)\frac{\mathrm{\pi }}{24}\\ \left(\text{D}\right)\frac{\mathrm{\pi }}{4}\end{array}$

Ans

$\begin{array}{l}\text{Let\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}={\int }_{0}^{\frac{2}{3}}\frac{\mathrm{dx}}{4+9{\mathrm{x}}^{2}}\\ \int \frac{1}{4+9{\mathrm{x}}^{2}}\text{\hspace{0.17em}}\mathrm{dx}=\int \frac{1}{{2}^{2}+{\left(3\mathrm{x}\right)}^{2}}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{2}{\mathrm{tan}}^{-1}\left(\frac{3\mathrm{x}}{2}\right)×\frac{1}{3}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}{\mathrm{tan}}^{-1}\left(\frac{3\mathrm{x}}{2}\right)=\mathrm{F}\left(\mathrm{x}\right)\\ \text{Therefore, by the second fundamental theorem, we have}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I}=\mathrm{F}\left(\frac{2}{3}\right)-\mathrm{F}\left(0\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}{\mathrm{tan}}^{-1}\left(\frac{3}{2}×\frac{2}{3}\right)-\frac{1}{6}{\mathrm{tan}}^{-1}\left(\frac{3}{2}×0\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}{\mathrm{tan}}^{-1}\left(1\right)-\frac{1}{6}{\mathrm{tan}}^{-1}\left(0\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}×\frac{\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{24}\\ \text{Hence, the correct option is C.}\end{array}$

## 1. Where can students find NCERT Solutions for Classes 6-12?

The following NCERT Solutions are available on the Extramarks website:

1. NCERT Solutions Class 12
2. NCERT Solutions Class 11
3. NCERT Solutions Class 10
4. NCERT Solutions Class 9
5. NCERT Solutions Class 8
6. NCERT Solutions Class 7
7. NCERT Solutions Class 6
8. NCERT Solutions Class 5
9. NCERT Solutions Class 4
10. NCERT Solutions Class 3
11. NCERT Solutions Class 2
12. NCERT Solutions Class 1

## 2. Are the solutions provided reliable for students to prepare for their board examinations?

Exercise 7.9 Class 12th Maths is challenging and the solutions for this are provided in the NCERT Solutions Class 12 Maths Chapter 7 Exercise 7.9. These learning resources from Extramarks strictly provide the solutions to the NCERT books. Before the solutions are released, they are reviewed by multiple people several times to ensure there are no incorrect answers. Although, some students have found other ways to solve a particular sum in a way that is different from the one provided in the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9. If the scope and method of solving a mathematical problem are correct, and the answer is right, the problem would be marked correct in the exams. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 provided by Extramarks are extremely helpful for students looking for extensive and efficient study materials, and if doubts persist even after referring to the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, then the students must engage in a good preparation plan for their examinations.