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 46.

Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 46.

Answers (1)

Answer :  \frac{8}{15}

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

\int_{0}^{\frac{\pi}{2}}\cos ^5x \; dx

Solution:

\begin{aligned} \int_{0}^{\pi / 2} \cos ^{5} x d x &=\int_{0}^{\pi / 2} \cos ^{4} x \cdot \cos x d x \\ &=\int_{0}^{\pi / 2}\left(\cos ^{2} x\right)^{2} \cos x d x \\ &=\int_{0}^{1 / 2}\left(1-\sin ^{2} x\right)^{2} \cos x d x \quad \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\therefore 1=\sin ^{2} \theta+\cos ^{2} \theta\right] \end{aligned}

Put  \sin x=t\Rightarrow \cos x \; dx=dt

when x=0  then t=0

whenx=\frac{\pi}{2}  then t=1

\begin{aligned} &\therefore \int_{0}^{\pi / 2} \cos ^{5} x d x=\int_{0}^{1}\left(1-t^{2}\right)^{2} d t \\ &\qquad \begin{aligned} & \int_{0}^{1}\left(1+t^{4}-2 t^{2}\right) d t \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right] \\ =& \int_{0}^{1} t^{0} d t+\int_{0}^{1} t^{4} d t-2 \int_{0}^{1} t^{2} d t \end{aligned} \end{aligned}

\begin{aligned} &=\left[\frac{t^{1+1}}{0+1}\right]_{0}^{1}+\left[\frac{t^{4+1}}{4+1}\right]_{0}^{1}-2\left[\frac{t^{2+1}}{2+1}\right]_{0}^{1} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad \because \int x^{n} d x=\frac{x^{n+1}}{x+1}+c \\ &=[t]_{0}^{1}+\frac{1}{5}\left[t^{5}\right]_{0}^{1}-\frac{2}{3}\left[t^{3}\right]_{0}^{1} \\ &=[1-0]+\frac{1}{5}\left[1^{5}-0^{5}\right]-\frac{2}{3}\left[1^{3}-0^{3}\right] \\ &=1+\frac{1}{5}-\frac{2}{3} \\ &=\frac{15+3-10}{15} \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 Supplementary Result 2026: Scanned copy, verification, re-evaluation dates announced
anam-khan

SC asks CBSE to protect students affected by OSM evaluation in admissions
anam-khan

CBSE Class 12 Compartment Result 2026 LIVE: 12th supplementary marksheet out at results.digilocker.gov.in

Latest Question