Question

Integrate xcos(5x)

Answer 1

I think that you have wrong u-substitution ( partial integration ):$\int {u} \, dv= uv- \int {v} \, du$
u=x,           dv=cos 5x dx
du=dx,       v=1/5 * sin 5x
Integral becomes:$= \frac{xsin 5x}{5}- \int { \frac{1}{5} sin 5x} \, dx= \\ = \frac{xsin5x}{5}- \frac{1}{5} (- \frac{1}{5}cos5x)= \frac{x sin5x}{5}+ \frac{cos5x}{25} +C$
Is it OK now? 

Answer 2

Hello,

$\int{x*cos(5x)} \, dx=x*\frac{sin(5x)}{5}- \frac{1}{5}*\int{sin 5x} \, dx\\=\frac{x*sin(5x)}{5}+\frac{cos(5x)}{25}+c$