Integration using parts

Question

I am struggling to evaluate this integral:
$$
\int e^{(\mu +r)t+\frac{r}{\alpha}\sin(\alpha t)}\cos(\alpha t)dt
$$
The integration just gets harder and harder when integrating by parts! Any tips?

Thanks in advance!

Answer 1

Hint: write $$e^{(\mu+r)t+\frac r\alpha\sin(\alpha t)}\cos(\alpha t)=e^{(\mu+r)t}e^{\frac r\alpha\sin(\alpha t)}\cos(\alpha t)$$Notice that the last two terms have trigonometric functions with the argument $(\alpha t)$. This indicates there may be a connection between these two terms. What happens if you integrate $e^{\frac r\alpha\sin(\alpha t)}\cos(\alpha t)$?