to avoid confusion, i changed u to xdo you mean $\displaystyle \int e^{ax} \sin nx~dx = \frac {e^{ax} (a \sin nx - n \cos nx)}{a^2 + n^2} + C$ ?
try integration by parts. let $\displaystyle u = \sin nx$ and $\displaystyle dv = e^{ax}~dx$. (i hope you remember what $\displaystyle u$ and $\displaystyle dv$ mean)
if you are really lazy, you can do a search for this question. it has been derived several times on this forum and many times on the internet, i'm sure, using both integration by parts and complex analysis
Hello,
Substitute : $\displaystyle \sin(x)=\frac{e^{ix}-e^{-ix}}{2i} \implies \sin(nx)=\frac{1}{2i} \cdot (e^{inx}-e^{-inx})$
$\displaystyle \int e^{ax} \sin(nx) dx=\frac{1}{2i} \cdot \left(\int e^{ax}e^{inx} dx-\int e^{ax}e^{-inx} dx\right) $$\displaystyle =\frac{1}{2i} \cdot \left(\int e^{x(a+in)} dx-\int e^{x(a-in)} dx \right)$
Consider i as a constant.