Question

Evaluate ${\int }_a^{b}x\sin xdx$

Answer 1

${\int }_a^{b}x\sin xdx$

Let $u=x\Rightarrow du=dx$

$v=\cos x\Rightarrow dv=\sin xdx$

Integrating by parts, we get

$={\left[x\cos x\right]}_a^b-{\int }_a^b\cos xdx$

$=[b\cos b-a\cos a]-{\left[\sin x\right]}_a^b$

$=[b\cos b-a\cos a]-[\sin b-\sin a]$

$=(b\cos b-\sin b)-(a\cos a-\sin a)$