Question

Find the integration of xsinx

Answer 1

We remember,

$\displaystyle\int fg'\ =\ fg-\int f'g$

Let,

$\leadsto\ f=x\quad\implies\quad f'=1\\\\\leadsto\ g'=\sin x\quad\implies\quad g=-\cos x$

So,

$\begin{aligned}&\int x\sin x\ dx\\\\\implies\ \ &-x\cos x-\int -\cos x\ dx\\\\\implies\ \ &\sin x-x\cos x\end{aligned}$