Need help with integration by parts I absolutely despise integration by parts, because it never seems to work for me. Here's an example: $$ \int 4x \sin(2x) \, \mathrm{d}x $$ What I did: $$ \int 4x \sin(2x) \, \mathrm{d}x = -2x \cos(2x) - \int - 4\cos(2x) \, \mathrm{d}x$$ Here I'm already stuck. I know about the ILATE/LIATE/whatever but it didn't work out for me either. What should I do at this juncture?

You've done the hard part of the integration by parts right; you're just off by a constant multiple. If we let $u = 4x,\; dv=\sin(2x)dx \implies du = 4\,dx,\;v=\frac{-\cos(2x)}{2}$.

When we put this together: $$\begin{align}uv - \int v\,du &= -2x\cos(2x) + \int\left(\frac{\cos(2x)}{2}\right)4\,dx\\\ &=-2x\cos(2x) + \int2\cos(2x)\,dx \end{align}$$ So, that's basically where you're at (except for the $2$ instead of the $4$). At this point, we can use $u$ substitution.

**Hint:**

> Let $u = 2x$.