Question

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

Answer 1

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

$={\int }_0^{\pi /4}xsinxsx-{\int }_0^{\pi /4}\left[x\right]sinxdx$

$=-xcos{\int }_0^{\pi /4}+sinx{\int }_0^{\pi /4}-0$

$=-\frac{\pi }{4}cos\frac{\pi }{4}+sin\frac{\pi }{4}$

$=\frac{1}{\sqrt{2}}-\frac{\pi }{4}\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}(1-\frac{\pi }{4})$ Ans