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 55.

Need solution for RD Sharma Maths Class 12 Chapter Definite Integrals exercise 19.2 question 55.
 

Answers (1)

Answer: \pi a

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

Given: \int_{-a}^{a}\sqrt{\frac{a-x}{a+x}}dx

Solution:

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

\begin{aligned} &=-2 a \int_{\frac{\pi}{2}}^{0}(1-\cos 2 \theta) d \theta \\ &=-2 a \int_{\frac{\pi}{2}}^{0} 1 d \theta+2 a \int_{\frac{\pi}{2}}^{0}(\cos 2 \theta) d \theta \\ &=-2 a[\theta]_{\frac{\pi}{2}}^{0}+2 a\left[\frac{\sin 2 \theta}{2}\right]_{\frac{\pi}{2}}^{0} \\ &=-2 a\left[0-\frac{\pi}{2}\right]+a\left[\sin 2 \times(0)-\sin 2 \times \frac{\pi}{2}\right] \\ &=-2 a\left(\frac{-\pi}{2}\right)+a[\sin 0-\sin \pi] \\ &=\pi a+a(0-0) \\ &=\pi a \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 Class 12 Board Exam 2026 LIVE: 12th Geography paper ends; set 1, 2 answer key, analysis
anam-khan

‘More exhausting than manual’: How CBSE on-screen marking system is draining teachers it was meant to help
anam-khan

CBSE Class 12 Accountancy Paper Analysis: Moderate overall; competency-based MCQs, no journal entry questions

Latest Question