Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Need solution for RD Sharma Maths Class 12 Chapter Definite Integrals exercise 19.2 question 54.

Need solution for RD Sharma Maths Class 12 Chapter Definite Integrals exercise 19.2 question 54.
 

Answers (1)

Answer: a^2\left [ \frac{\pi}{4}-\frac{1}{2} \right ]  

Hint: Use indefinite formula and the limits to solve this integral

Given: \int_{0}^{a} x \sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}} d x

Solution:

\int_{0}^{a} x \sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}} d x
Put x^2=a^2\cos 2\theta \Rightarrow 2x dx=a^2-\sin2\theta .2d\theta
\Rightarrow dx=-a^2\sin 2\theta x\; d\theta
When x=0 then \theta =\frac{\pi}{4} & when x=a then \theta =0
\begin{aligned} &\therefore \int_{0}^{a} x \sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}} d x=\int_{\frac{\pi}{4}}^{0} \sqrt{\frac{a^{2}(1-\cos 2 \theta)}{a^{2}(1+\cos 2 \theta)}}\left(-a^{2} \sin 2 \theta\right) d x \\ &=-a^{2} \int_{\frac{\pi}{4}}^{0} \sqrt{\frac{a^{2} 2 \sin ^{2} \theta}{a^{2} 2 \cos ^{2} \theta}}(\sin 2 \theta) d x \\ &=-a^{2} \int_{\frac{\pi}{4}}^{0} \frac{\sin \theta}{\cos \theta}(2 \sin \theta \cos \theta) d \theta \mid \end{aligned}

\begin{aligned} &=-a^{2} \int_{\frac{\pi}{4}}^{0}\left(2 \sin ^{2} \theta\right) d \theta \\ &=-a^{2} \int_{\frac{\pi}{4}}^{0}(1-\cos 2 \theta) d \theta \\ &=-a^{2} \int_{\frac{\pi}{4}}^{0} 1 d \theta+a^{2} \int_{\frac{\pi}{4}}^{0}(\cos 2 \theta) d \theta \\ &=-a^{2}[\theta]_{\frac{\pi}{4}}^{0}+a^{2}\left[\frac{\sin 2 \theta}{2}\right]_{\frac{\pi}{4}}^{0} \\ &=-a^{2}\left[0-\frac{\pi}{4}\right]+\frac{a^{2}}{2}\left[\sin 2(0)-\sin 2 \times \frac{\pi}{4}\right] \\ &=\frac{a^{2} \pi}{4}-\frac{a^{2}}{2} \cdot 1 \\ &=a^{2}\left[\frac{\pi}{4}-\frac{1}{2}\right] \end{aligned}

 

Posted by

infoexpert24

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Date Sheet 2027 LIVE: Class 10, 12 board exam schedule soon; passing marks, paper pattern
anam-khan

CBSE Class 12 West Asia special exam date sheet 2026 out; admit cards from October 12
anam-khan

CBSE Class 10, 12 private exams 2026-27 registration begins on cbse.gov.in, check eligibility and fees

Latest Question