Does $\int _0^{\pi }e^x\sin ^n x\:\mathrm{d}x$ have a closed form

Question

Does a closed form for $\displaystyle\int _0^{\pi }e^x\sin ^n x\:\mathrm{d}x$ exist?

I tried to evaluate this with values like $n=1,2$ with integration by parts and it seemed fine but when i tried with higher values such as $n=3,4,5$ it became more tedious and couldn't manage to evaluate.

Could you please help me find the closed form of this expression please?.

Answer 1

Let $I_n = \int _0^{\pi }e^x\sin ^n x dx$ and integrate by parts twice to get the recursive equation below

$$I_n = \frac{n(n-1)}{n^2+1}I_{n-2}$$

with $I_0= e^\pi-1$ and $I_1=\frac12(1+e^\pi)$.