Re: integration by parts.

It looks to me that you have correctly applied the method, although don't forget the constant of integration.

If you have doubts about your indefinite integration result, recall that you may use differentiation as a check:

$\displaystyle \frac{d}{dx}\left(\frac{5x}{4}\sin(4x)+\frac{5}{16 }\cos(4x)+C\right)=\frac{5x}{4}(4\cos(4x))+\frac{5 }{4}\sin(4x)-\frac{5}{4}\sin(4x)+0=5x\cos(4x)$

Since the derivative of the anti-derivative is the original integrand, you know the result is correct.