Question

qual a integral de x sen (5x) dx

Answer 1

Pela integração por partes:
$\displaystyle \int udv=vu-\int vdu$
note que:
$dv=vdx\\du=udx$
então na função acima definimos u, du, v e dv:
$u=x\\du=dx\\v=-\frac{1}{5}\cos\,5x\\dv=\sin 5x\,dx$
logo:
$\displaystyle i)~~~~\int x\sin5x\,dx=-\frac{1}{5}x\cos\,5x-\int-\frac{1}{5}\cos 5x\,dx\\\\ii)~~\int x\sin5x\,dx=-\frac{1}{5}x\cos5x+\int\frac{1}{5}\cos\,5x\,dx\\\\iii)\int\frac{1}{5}\cos5x\,dx\implies 5x=u\implies dx=\frac{du}{5}\\\\iv)\int\frac{1}{25}\cos\,u\,du=\frac{1}{25}\sin\,u=\boxed{\frac{1}{25}\sin5x}\\\\v)\int x\sin5xdx=\boxed{\frac{1}{25}\sin5x-\frac{1}{5}\cos\,5x+C}$