Evaluate the Integral: $\int e^{2\theta}\ \sin 3\theta\ d\theta$ $\int e^{2\theta}\ \sin 3\theta\ d\theta$
After Integrating by parts a second time, It seems that the problem will repeat for ever. Am I doing something wrong. I would love for someone to show me using the method I am using in a clean and clear fashion. Thanks. 

When you do it with integration by parts, you have to go in the "same direction" both times. For instance, if you initially differentiate $e^{2 \theta}$, then you need to differentiate $e^{2 \theta}$ again; if you integrate it, you will wind up back where you started. If you do this, you should find something of the form
$$\int e^{2 \theta} \cos(3 \theta) d \theta = f(\theta) + C \int e^{2 \theta} \cos(3 \theta) d \theta$$
where $C$ is not $1$. Therefore you can solve the equation for the desired quantity:
$$\int e^{2 \theta} \cos(3 \theta) d \theta = \frac{f(\theta)}{1-C}.$$
There is also a nice approach with complex numbers: $\cos(3 \theta)=\frac{e^{3 i \theta}+e^{-3 i \theta}}{2}$, so your integral is
$$\frac{1}{2} \int e^{(2+3i) \theta} + e^{(2-3i) \theta} d \theta$$
which are pretty easy integrals. You do some complex number arithmetic and it works out.
