NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.9) Exercise 7.9
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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.
Mathematics plays a vital role in everyday life as a result of its ability to enhance the application of logical knowledge, a skill that underscores the importance of mathematics and various mathematical sciences. Through the practical knowledge that comprises mathematics, and its associated sciences, the uninterrupted operation of a diverse range of processes has been made possible.
Class 12 students should inculcate the habit of solving past years’ papers and sample question papers. This helps them with error correction and acquainting themselves with the variety of questions that can appear in the exam. The Extramarks website is a repository of various interactive and engaging learning tools that can help students selfassess their level of preparation and prepare efficiently for their examinations.
It is therefore essential that students learn how to apply their retained knowledge of theorems, concepts, and formulae to carry out complex calculations in a systematic and organized manner. For those who have chosen science as their academic stream following the completion of their Class 10 boards, Mathematics is one of their core subjects. Mathematics has its concepts and ideas reinstated in subjects like physics and chemistry. Subjects like physics and chemistry are incomplete without a nascent knowledge of mathematics; therefore, teachers implore the students to focus greatly on mathematics.
According to educators across the country, the major challenge that students face is the complexity of the curriculum. As students proceed with the syllabus, the doubts they have keep piling up, and it becomes burdensome for the teacher to address this later on. Therefore, when doubts arise while a student is solving the problems, they can attend to the doubts by themselves with the help of the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9.
Extramarks, therefore, makes available the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 to assist students regarding this problem. One of the major portions of the mathematics syllabus is Integrals and Extramarks, makes accessible the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, which is a collection of all the solutions to Chapter 7 Exercise 7.9 NCERT in one place. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, provides explanations of complex steps provided by qualified professionals in the field. Students have claimed to have greatly benefited from the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9.
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.
Extramarks has observed that the easy solution to the problem is to have a tangible and easily accessible learning resource for all the NCERT questions. Therefore, the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 is one of the most convenient ways students can seek help if they face a doubt. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 are curated by professionals who are aware of the recent trends in the CBSE syllabus and its patterns. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, select the most discernible method to solve that can be understood by students regardless of their academic prowess. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, are reviewed multiple times, and they are only made available on the Extramarks website after it is ensured that the solutions are completely credible.
The NCERT Solutions for Class 12 Maths Chapter 7 Exercise, 7.9, have helped innumerable students who face difficulties with the syllabus and the difficulty of the CBSE exams. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 are extremely beneficial for students because a student is always searching for help, and they can refer to the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 according to their convenience.
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
The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 helps students and teachers greatly in revision and assessment. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, provide a student with a lot of practise resources. A Class 12 CBSE student can be expected to be busy with revising all the core science subjects, and doing the reference work for all subjects can be very difficult for them to balance. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 aid students by providing them with a chance to clear their doubts themselves because all the solutions provided in the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9 are widely accepted to be the simplest way to solve a particular problem. The solutions provided in the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, sometimes become complicated because they sometimes use formulas and references from other chapters, and students generally feel very confused while following the long and extensive processes, but the answers provided are curated in a way that is easily comprehended by students. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, provide solutions as well as explanations for optimum assistance. Therefore, when doubt persists even after referring to the solutions, it can be a sign that there is some incoherence in the students’ understanding of a concept, and they must address the topic immediately.
Resources like the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.9, are extremely helpful for students because, given the tight schedule that they follow, it is extremely difficult to maintain a perfect balance among all the subjects. Most of these students are preparing for other competitive exams, and having the entire solutions approved by pioneers in the respective field gives the student a muchneeded confidence boost. Students can revise any subject according to their convenience, and they can do it fearlessly because they are within easy reach of help and support.
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.
Extramarks have other resources for other subjects and other exercises as well, and they can be easily accessed by any student from either their website or their mobile app. The Extramarks website is consistently chosen as an appropriate platform to access learning tools, and students have shared positive feedback on the solutions.
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
$\begin{array}{l}\text{Let\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}={\int}_{0}^{\frac{2}{3}}\frac{\mathrm{dx}}{4+9{\mathrm{x}}^{2}}\\ \int \frac{1}{4+9{\mathrm{x}}^{2}}\text{\hspace{0.17em}}\mathrm{dx}=\int \frac{1}{{2}^{2}+{\left(3\mathrm{x}\right)}^{2}}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{2}{\mathrm{tan}}^{1}\left(\frac{3\mathrm{x}}{2}\right)\times \frac{1}{3}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}{\mathrm{tan}}^{1}\left(\frac{3\mathrm{x}}{2}\right)=\mathrm{F}\left(\mathrm{x}\right)\\ \text{Therefore, by the second fundamental theorem, we have}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I}=\mathrm{F}\left(\frac{2}{3}\right)\mathrm{F}\left(0\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}{\mathrm{tan}}^{1}(\frac{3}{2}\times \frac{2}{3})\frac{1}{6}{\mathrm{tan}}^{1}(\frac{3}{2}\times 0)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}{\mathrm{tan}}^{1}\left(1\right)\frac{1}{6}{\mathrm{tan}}^{1}\left(0\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{6}\times \frac{\mathrm{\pi}}{4}=\frac{\mathrm{\pi}}{24}\\ \text{Hence, the correct option is C.}\end{array}$
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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.