Question

The value of the integral $\int\limits_{-\pi / 2}^{\pi / 2}\left(x^{2}+\ln \frac{\pi+x}{\pi-x}\right) \cos x d x$ is

• A. 0
• B. $\frac{\pi^{2}}{2}-4$
• C. $\frac{\pi^{2}}{2}+4$
• D. $\frac{\pi^{2}}{2}$

Answer 1

Answer: (B)

$\int\limits_{-\pi / 2}^{\pi / 2}\left\{x^{2}+\ln \left(\frac{\pi+x}{\pi-x}\right)\right\} \cos x d x$
$=\int\limits_{-\pi / 2}^{\pi / 2} x^{2} \cos x d x+\int\limits_{-\pi / 2}^{\pi / 2} \ln \left(\frac{\pi+x}{\pi-x}\right) \cos x d x$
$=2 \int\limits_{0}^{\pi / 2} x^{2} \cos x d x$
$=2\left[x^{2} \sin x+2 x \cos x-2 \sin x\right]_{0}^{\pi / 2}$
$ = 2 \left[\frac{\pi^2}{4} - 2\right] = \frac{\pi^2}{2} -4$.