Question

${\int }_0^{\pi /4}\sin xd(x-[x\left]\right)$is equal to

• A. $\frac{1}{2}$
• B. $1-\frac{1}{\sqrt{2}}$
• C. $1$
• D. none of these

Answer 1

Answer: (B)

$1-\frac{1}{\sqrt{2}}$

${\int }_0^{\pi /4}\sin xd(x-[x\left]\right)$

$={\int }_0^{\pi /4}\sin xdx-{\int }_0^{\pi /4}\sin xd\left[x\right]$
$={\int }_0^{\pi /4}\sin xdx-0$

$={[-\cos x]}_0^{\pi /4}=1-\frac{1}{\sqrt{2}}$