Question

$\underset{0}{\overset{\pi /2}{\int }}(x-[\cos x\left]\right)dx=$ _____
(where $\left[t\right]=$ greatest integer less or equal to $t$)

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

Answer 1

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

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