Question

Solve : $2\underset{0}{\stackrel{\pi /2}{\int }}xsinxdx$

Answer 1

$I=2{\int }_0^{\frac{\pi }{2}}x\sin xdx$

Integration by parts

${\int }_a^{b}f\left(x\right)g\left(x\right)dx=f\left(x\right)\int g\left(x\right)dx{\int }_a^b-{\int }_a^b{f}^{\prime }\left(x\right)g\left(x\right)dx$

$I=2\left[x\right[-\cos x{]}_0^{\frac{\pi }{2}}-{\int }_0^{\frac{\pi }{2}}\left(1\right)(-\cos x)dx]$

$=2[-(\frac{\pi }{2}\cos \frac{\pi }{2}-0)+{\int }_0^{\frac{\pi }{2}}\cos xdx]$

$=2\left[{\left[\right(0-0)+\sin x]}_0^{\frac{\pi }{2}}\right]$

$=2[\sin \frac{\pi }{2}-\sin (0\left)\right]\Rightarrow 2[1-0]\Rightarrow 2$

$\therefore 2{\int }_0^{\frac{\pi }{2}}x\sin xdx=2$.