NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.11) Exercise 7.11
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NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.11) Exercise 7.11
Exercise 7.11 Maths Class 12 deals with the chapter on Integrals. Students should use Ex 7.11 Class 12 Maths NCERT Solutions to help them make the task of learning and revising the chapter easier. NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 with all the other exercises of the chapter are available for download in one place. The NCERT Solutions Class 12 Maths Chapter 7 Exercise 7.11 are prepared by an expert teacher at Extramarks by following the NCERT (CBSE) book guidelines. NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 contains questions and solutions to the exercise to assist students in revising the entire syllabus and scoring higher in the upcoming exams.
Access the NCERT Solution for Class 12 Maths Chapter 7 – Integrals
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.11
Choosing the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 by Extramarks is the goto option for students. Chapter 7 of Class 12, Integrals, has a lot of exercises for students to deal with. This page provides the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 in PDF format for ease of access. Students should be able to download this via the Extramarks’ website. NCERT Solutions for all the other exercises of the chapter can also be found easily on the website under the same category.
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Q.1
$\begin{array}{l}\text{Show that}{\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\text{\hspace{0.17em}}\mathrm{dx}=2{\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx},\text{\hspace{0.17em}if f and g are defined}\\ \text{as}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\mathrm{a}\mathrm{x})\mathrm{}\text{and}\mathrm{g}\left(\mathrm{x}\right)+\mathrm{g}(\mathrm{a}\mathrm{x})=4\end{array}$
Ans
$\begin{array}{l}\text{Let\hspace{0.17em}\hspace{0.17em}}\mathrm{I}={\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\text{\hspace{0.17em}}\mathrm{dx}...\left(\mathrm{i}\right)\\ \text{\hspace{0.17em} \hspace{0.17em}}={\int}_{0}^{\mathrm{a}}\mathrm{f}(\mathrm{a}\mathrm{x})\mathrm{g}(\mathrm{a}\mathrm{x})\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}}[\xe2\u02c6\mu \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int}_{0}^{\mathrm{a}}\mathrm{f}(\mathrm{a}\mathrm{x})\mathrm{dx}]\\ \text{\hspace{0.17em} \hspace{0.17em}}={\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}(\mathrm{a}\mathrm{x})\text{\hspace{0.17em}}\mathrm{dx}\text{\hspace{0.17em}\hspace{0.17em}}...\left(\mathrm{ii}\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}[\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\mathrm{a}\mathrm{x})]\\ \text{Adding equation}\left(\text{i}\right)\text{and}\left(\text{ii}\right)\text{, we get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2I}={\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\text{\hspace{0.17em}}\mathrm{dx}+{\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}(\mathrm{a}\mathrm{x})\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}}={\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\{\mathrm{g}\left(\mathrm{x}\right)+\mathrm{g}(\mathrm{a}\mathrm{x})\}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\left(4\right)\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}}[\xe2\u02c6\mu \text{\hspace{0.17em}\hspace{0.17em}}\mathrm{g}\left(\mathrm{x}\right)+\mathrm{g}(\mathrm{a}\mathrm{x})=4]\\ \text{\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}=2{\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\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}Hence\xe2\u20ac\u2039 proved.}\end{array}$
Q.2
$\begin{array}{l}\mathrm{Choose}\text{}\mathrm{the}\text{}\mathrm{correct}\text{}\mathrm{answer}\\ \text{The value of}{\int}_{\frac{\text{\pi}}{2}}^{\frac{\text{\pi}}{2}}\left({\text{x}}^{\text{3}}\text{+xcosx}+{\mathrm{tan}}^{5}\text{x}+1\right)\mathrm{dx}\text{}\mathrm{is}\\ \left(\mathrm{A}\right)\text{}0\\ \left(\mathrm{B}\right)\text{}2\\ \left(\mathrm{C}\right)\text{}\mathrm{\pi}\\ \left(\mathrm{D}\right)\text{}1\end{array}$
Ans
$\begin{array}{l}\text{Let\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}={\int}_{\frac{\mathrm{\pi}}{2}}^{\frac{\mathrm{\pi}}{2}}({\mathrm{x}}^{3}+\mathrm{xcosx}+{\mathrm{tan}}^{5}\mathrm{x}+1)\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}={\int}_{\frac{\mathrm{\pi}}{2}}^{\frac{\mathrm{\pi}}{2}}{\mathrm{x}}^{3}\mathrm{dx}+{\int}_{\frac{\mathrm{\pi}}{2}}^{\frac{\mathrm{\pi}}{2}}\mathrm{xcosxdx}+{\int}_{\frac{\mathrm{\pi}}{2}}^{\frac{\mathrm{\pi}}{2}}{\mathrm{tan}}^{5}\mathrm{xdx}+{\int}_{\frac{\mathrm{\pi}}{2}}^{\frac{\mathrm{\pi}}{2}}1\mathrm{dx}\\ \text{Since},\text{if f}\left(\mathrm{x}\right)\text{is even function then,}{\int}_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\text{\hspace{0.17em}}\mathrm{dx}=2{\int}_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\text{\hspace{0.17em}}\mathrm{dx}\\ \text{and if f}\left(\text{x}\right)\text{is odd function then, \hspace{0.17em}\hspace{0.17em}}{\int}_{\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\text{\hspace{0.17em}}\mathrm{dx}=0\\ \therefore \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}=0+0+0+2{\int}_{0}^{\frac{\mathrm{\pi}}{2}}1\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}=2{{\left[\mathrm{x}\right]}_{0}}^{\frac{\mathrm{\pi}}{2}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}=2\left(\frac{\mathrm{\pi}}{2}\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}=\mathrm{\pi}\\ \text{Therefore, the correct option is C.}\end{array}$
Q.3
$\begin{array}{l}\mathrm{Choose}\text{}\mathrm{the}\text{}\mathrm{correct}\text{}\mathrm{answer}\\ \text{The value of}{\int}_{0}^{\frac{\mathrm{\pi}}{2}}\text{log}\left(\frac{\text{4+3sinx}}{\text{4+3cosx}}\right)\text{dx is}\\ \left(\mathrm{A}\right)\text{}2\\ \left(\mathrm{B}\right)\text{}\frac{3}{4}\\ \left(\mathrm{C}\right)\text{}0\\ \left(\mathrm{D}\right)\text{}2\end{array}$
Ans
$\begin{array}{l}\text{Let\hspace{0.17em}\hspace{0.17em}}\mathrm{I}={\int}_{0}^{\frac{\mathrm{\pi}}{2}}\mathrm{log}\left(\frac{4+3\mathrm{sinx}}{4+3\mathrm{cosx}}\right)\text{\hspace{0.17em}}\mathrm{dx}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}...\left(\mathrm{i}\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}={\int}_{0}^{\frac{\mathrm{\pi}}{2}}\mathrm{log}\left\{\frac{4+3\mathrm{sin}(\frac{\mathrm{\pi}}{2}\mathrm{x})}{4+3\mathrm{cos}(\frac{\mathrm{\pi}}{2}\mathrm{x})}\right\}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={\int}_{0}^{\frac{\mathrm{\pi}}{2}}\mathrm{log}\left(\frac{4+3\mathrm{cosx}}{4+3\mathrm{sinx}}\right)\text{\hspace{0.17em}}\mathrm{dx}\text{\hspace{0.17em}}...\left(\mathrm{ii}\right)\\ \text{Adding equation}\left(\text{i}\right)\text{and equation}\left(\text{ii}\right)\text{, we get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2I}={\int}_{0}^{\frac{\mathrm{\pi}}{2}}\{\mathrm{log}\left(\frac{4+3\mathrm{sinx}}{4+3\mathrm{cosx}}\right)+\mathrm{log}\left(\frac{4+3\mathrm{cosx}}{4+3\mathrm{sinx}}\right)\}\text{\hspace{0.17em}}\mathrm{dx}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}={\int}_{0}^{\frac{\mathrm{\pi}}{2}}\left\{\mathrm{log}(\frac{4+3\mathrm{sinx}}{4+3\mathrm{cosx}}\times \frac{4+3\mathrm{cosx}}{4+3\mathrm{sinx}})\right\}\text{\hspace{0.17em}}\mathrm{dx}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={\int}_{0}^{\frac{\mathrm{\pi}}{2}}\mathrm{log}\left(1\right)\text{\hspace{0.17em}}\mathrm{dx}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{I}=0\\ \text{Therefore, the correction option is C.}\end{array}$
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1. How many questions are there in the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11?
The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 solutions by Extramarks has all the 21 questions and solutions of Exercise 7.11 Maths Class 12.
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The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 can be used as a guide while solving the chapter. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 can also come in handy during the lastminute preparation and revision before the examination. Class 12 Mathematics can be tough; therefore, it is good to have something like the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 by the student’s side to help them when they are having trouble.This will also help to keep the flow going while working on Exercise 7.11 Math Class 12.
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The foremost preparatory step that a student needs to take is to study and analyse the syllabus properly. Then devise a plan according to it. Using past years’ papers and study material like the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 can prove to be extremely useful. Getting command over weak areas and continuously revising the strong areas of one’s preparation is also important. Selfassessment by using NCERT Solutions like NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 and other revision material also enhances preparation and, in turn, results.
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The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 contain detailed solutions to the chapter on Integrals. Extramarks makes sure that students gain clear knowledge and achieve higher marks in their forthcoming examinations. The NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11 not only have the solutions but also proper steps explaining everything needed by the student.
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