Question

The value of the integral ${\int }_0^{\pi }xco{s}^{2}xdx$ is

• A. ${\pi }^2/8$
• B. ${\pi }^2/4$
• C. ${\pi }^2/2$
• D. ${\pi }^2$

Answer 1

The correct option is B ${\pi }^2/4$

$I={\int }_0^{\pi }xco{s}^{2}xdx={\int }_0^{\pi }(\pi -x)co{s}^2(\pi -x)dx$
(By definite integral property)
$I=\pi {\int }_0^{\pi }co{s}^{2}xdx-{\int }_0^{\pi }xco{s}^{2}xdx$
$2I=\pi {\int }_0^{\pi }co{s}^{2}xdx=2\pi {\int }_0^{\pi /2}co{s}^{2}xdx$