Q:

Justifying the integral identity “$\int \mathrm{e}^{cx} \sin bx\, \mathrm{d}x = \frac{\mathrm{e}^{cx}}{b^2 + c^2}(c \sin bx - b \cos bx)$”

I suck at integrals.  Here on this Wiki list, it says that
$$\int \mathrm{e}^{cx} \sin bx \,\mathrm{d}x = \frac{\mathrm{e}^{cx}}{b^2 + c^2}(c \sin bx - b \cos bx),$$
which is easy to check – but how could I have derived that? 

A:

Just write $\sin(bx)$ as $\text{Im}\left(e^{ibx}\right)$, so that the LHS is just the imaginary part of $\frac{1}{c+ib} e^{(c+ib)x}$.

