Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 44.

Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 44.

Answers (1)

Answer  : \frac{\pi}{3}

Hint   : use indefinite formula and the limit to solve this integral

Given   : \int_{0}^{(\Pi)^{\frac{2}{3}}} \sqrt{x} \cos ^{2} x^{\frac{3}{2}} d x

Solution  :  \int_{0}^{(\Pi)^{\frac{2}{3}}} \sqrt{x} \cos ^{2} x^{\frac{3}{2}} d x

put x^{\frac{3}{2}}=t \Rightarrow \frac{3}{2} x^{\frac{3}{2}-1} d x=d t \Rightarrow \frac{3}{2} x^{\frac{1}{2}} d x=d t \Rightarrow d x=\frac{2}{3 \sqrt{x}} d t

when x=0 then t=0 when x=(\Pi)^{\frac{2}{3}} then t=\Pi

Therefore

 

\int_{0}^{(\Pi)^{\frac{2}{3}}} \sqrt{x} \cos ^{2} x^{\frac{3}{2}} d x

\begin{aligned} &=\int_{0}^{\pi} \sqrt{x} \cos ^{2} t \frac{2}{3 \sqrt{x}} d t \\ &=\frac{2}{3} \int_{0}^{\pi} \cos ^{2} t d t \\ &=\frac{2}{3} \int_{0}^{\pi}\left(\frac{1+\cos 2 t}{2}\right) d t \\ &=\frac{1}{3} \int_{0}^{\pi}(1+\cos 2 t) d t \\ &=\frac{1}{3}\left[\int_{0}^{\pi} 1 d t\right]+\frac{1}{3}\left[\int_{0}^{\pi} \cos 2 t d t\right] \\ &=\frac{1}{3}[t]_{0}{ }^{\pi}+\frac{1}{3}\left[\frac{\sin 2 t}{2}\right]_{0}^{\pi} \\ &=\frac{1}{3} \Pi-0+\frac{1}{6}[\sin 2 \pi-\sin 2 \times 0] \end{aligned}

=\frac{1}{3}\Pi+\frac{1}{6}(0-0)

=\frac{\Pi}{3}

 

 

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 extends Class 12 re-evaluation, verification deadline to June 7 amid portal issues
anam-khan

CBSE approaches Delhi Police over coordinated cyber attacks on post-result services portal
anam-khan

'CBSE made to adopt OSM system at inflated rates': Congress alleges, demands Pradhan's resignation

Latest Question