In general, if you want to find $$ \int e^{ax}\cdot \sin{bx}\cdot dx$$ you can argue as follows:
Note that for any $\alpha$ or $\beta$, you have
$$\eqalign{
& \frac{d}{{dx}}\left( {{e^{\alpha x}}\sin \beta x} \right) = \alpha {e^{\alpha x}}\sin \beta x + \beta {e^{\alpha x}}\cos \beta x \cr
& \frac{d}{{dx}}\left( {{e^{\alpha x}}\cos \beta x} \right) = \alpha {e^{\alpha x}}\cos \beta x - \beta {e^{\alpha x}}\sin \beta x \cr} $$
so that any integral of the form
$$ \int e^{\alpha x}\cdot \sin{\beta x}\cdot dx$$
is a linear combination of the former functions. Let's then find $c_1$ and $c_2$ such that
$$\frac{d}{{dx}}\left( {{c_1}{e^{\alpha x}}\sin \beta x + {c_2}{e^{\alpha x}}\cos \beta x} \right) = {e^{\alpha x}}\sin \beta x$$
$${c_1}\alpha {e^{\alpha x}}\sin \beta x + {c_1}\beta {e^{\alpha x}}\cos \beta x + {c_2}\alpha {e^{\alpha x}}\cos \beta x - {c_2}\beta {e^{\alpha x}}\sin \beta x = {e^{\alpha x}}\sin \beta x$$
This means we need
$$\eqalign{
& {c_1}\alpha - {c_2}\beta = 1 \cr
& {c_1}\beta + {c_2}\alpha = 0 \cr} $$
This will yield with little work
$$\eqalign{
& {c_1} = \frac{\alpha }{{{\alpha ^2} + {\beta ^2}}} \cr
& {c_2} = - \frac{\beta }{{{\alpha ^2} + {\beta ^2}}} \cr} $$
which means that, in general:
$$\int {{e^{\alpha x}}} \cdot\sin \beta x\cdot dx = {e^{\alpha x}}\frac{{\alpha \sin \beta x - \beta \cos \beta x}}{{{\alpha ^2} + {\beta ^2}}} + C$$
Analogously, you will get that
$$\int {{e^{\alpha x}}} \cdot\cos \beta x\cdot dx = {e^{\alpha x}}\frac{{\alpha \cos \beta x + \beta \sin \beta x}}{{{\alpha ^2} + {\beta ^2}}} + C$$
Hope this helps!