Question

$\underset{0}{\overset{\pi /2}{\int }}(x-[\sin x\left]\right)dx=$ _____ , where $\left[x\right]$ is the greatest integer function.

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

Answer 1

The correct option is D $\frac{{\pi }^2}{8}$

$\left[\sin x\right]=0,\forall x\in (0,\frac{\pi }{2})$
$\therefore \underset{0}{\overset{\pi /2}{\int }}(x-[\sin x\left]\right)dx$
$=\underset{0}{\overset{\pi /2}{\int }}xdx$
$={\left[\frac{x^2}{2}\right]}_0^{\pi /2}$
$=\frac{{\pi }^2}{8}$