Question

$\underset{-2}{\overset{2}{\int }}|x\cos \pi x|dx$ is equal to

• A. $\frac{8}{\pi }$
• B. $\frac{4}{\pi }$
• C. $\frac{2}{\pi }$
• D. $\frac{1}{\pi }$

Answer 1

Answer: (A)

$I=\underset{-2}{\overset{2}{\int }}\left|x\cos \pi x\right|dx$
$=2\underset{0}{\overset{2}{\int }}\left|x\cos \pi x\right|dx$
$=2\left[\underset{0}{\overset{1/2}{\int }}\left(x\cos \pi x\right)dx-\underset{1/2}{\overset{3/2}{\int }}\left(x\cos \pi x\right)dx+\underset{3/2}{\overset{2}{\int }}\left(x\cos \pi x\right)dx\right]$
Now $\int x\cos \pi xdx=\frac{x\sin \pi x}{\pi }+\frac{\cos \pi x}{{\pi }^2}$
$\therefore I=2\left[\left(\frac{1}{2\pi }-\frac{1}{{\pi }^2}\right)-\left(-\frac{3}{2\pi }-\frac{1}{2\pi }\right)+\left(\frac{1}{{\pi }^2}+\frac{3}{2\pi }\right)\right]$
$=2\left[\frac{1}{2\pi }-\frac{1}{{\pi }^2}+\frac{3}{2\pi }+\frac{1}{2\pi }+\frac{1}{{\pi }^2}+\frac{3}{2\pi }\right]$
$=2\left(\frac{8}{2\pi }\right)=\frac{8}{\pi }$