Recursive integration by parts I'm trying to integrate the product of an exponential and trigonometric function using integration by parts.
However, after 2 loops I realized that it would keep looping forever.
Is there any way to integrate this function?
$$\int_0^\pi e^{2t}\sin t dt$$
This are my attempt (I hope the picture and my writing is clear enough to be read): 


First, expand the integral using integration by parts until it reappears:
$$\begin{align*}
\int e^{2t} \sin{t} \ dt & ~=~ \frac{1}{2} e^{2t} \sin{t} - \frac{1}{2} \int e^{2t} \cos{t} \ dt \\
& ~=~ \frac{1}{2} e^{2t}\sin{t} - \frac{1}{4} e^{2t}\cos{t} - \frac{1}{4} \int e^{2t} \sin{t} \ dt
\end{align*}$$
Now combine like terms and multiply by $4 / 5$ to obtain:
$$\begin{align*}
\int e^{2t} \sin{t} \ dt & ~=~ \frac{2}{5} e^{2t}\sin{t} - \frac{1}{5} e^{2t} \cos{t} \\
\end{align*}$$
