Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 40 maths textbook solution.

Explain solution for  RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 40 maths textbook solution.

Answers (1)

Answer:-  \frac{1}{\pi}\left(\frac{5}{2}-\frac{1}{\pi}\right)

Hints:-  You must know the integral rules of trignometric functions.

Given:-  \int_{0}^{3 / 2}|x \cos \pi x| d x

Solution : \int_{0}^{3 / 2}|x \cos \pi x| d x

               \begin{aligned} &0<x<\frac{1}{2} \\ &0<\pi x<\frac{\pi}{2} \\ &\cos \pi x>0 \rightarrow x \cos (\pi x)>0 \end{aligned}

               \begin{aligned} &|x \cos \pi x|=x \cos \pi x \\ &\frac{1}{2}<x<\frac{3}{2} \end{aligned}

                 \begin{aligned} &\frac{\pi}{2}<\pi x<\frac{3 \pi}{2} \\ &\cos \pi x<0 \rightarrow x \cos \pi x<0 \\ &|x \cos \pi x|=-x \cos \pi x \end{aligned}

                \begin{aligned} &I=\int_{0}^{3 / 2}|x \cos \pi x| \cdot d x \\ &I=\int_{0}^{3 / 2} x \cos \pi x+\int_{1 / 2}^{3 / 2}-(x \cos \pi x) \\ &I=\int_{0}^{1 / 2} x \cos \pi x-\int_{1 / 2}^{3 / 2} x \cos \pi x \cdot d x \end{aligned}

               \begin{aligned} &\int x \cos \pi x=x \frac{\sin \pi x}{\pi}-\int \frac{\sin \pi x}{\pi} \\ &\int x \cos \pi x=\frac{x}{\pi} \sin \pi x+\frac{\cos \pi x}{\pi^{2}} \end{aligned}

              \begin{aligned} &I=\left[\frac{x}{\pi} \sin \pi x+\frac{\cos \pi x}{\pi^{2}}\right]_{0}^{\frac{1}{2}}-\left[\frac{x}{\pi} \sin \pi x+\frac{\cos \pi x}{\pi^{2}}\right]_{\frac{1}{2}}^{3 / 2} \\ &I=\left[\frac{1}{\pi}\left(\frac{1}{2}-0\right)+\frac{1}{\pi^{2}}(0-1)\right]-\left[\frac{1}{\pi}\left(\frac{3}{2}(-1)\right)+\frac{1}{\pi^{2}}(0-0)\right] \end{aligned}

              \begin{aligned} &=\left[\frac{1}{2 \pi}-\frac{1}{\pi^{2}}\right]-\left[\frac{-2}{\pi}\right] \\ &=\left[\frac{5}{2 \pi}-\frac{1}{\pi^{2}}\right] \\ &=\frac{1}{\pi}\left[\frac{5}{2}-\frac{1}{\pi}\right] \end{aligned}

           

Posted by

infoexpert23

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 12th Result 2026: Board refutes OSM glitch claims, says Class 12 results likely in third week of May
anam-khan

CBSE 12th Result 2026 LIVE: Class 12 results soon at cbse.gov.in, direct link, digilocker, pass percentage
anam-khan

CBSE 12th Result 2026 Date LIVE: Class 12 marksheet on cbse.gov.in, DigiLocker soon; OSM system, passing marks

Latest Question