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 introduces CCTV policy in board exams 2025; schools fixed as centres must have facility
anam-khan

CBSE gives parents, schools till October 16 to register students in Classes 9, 11
anam-khan

CBSE Board Exam 2025: Class 9, 11 registration begins today; fee details, guidelines

Latest Question