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. 

Here is a plain answer for your reference:
Let $\simeq$ denote the equality sign up to a constant. 
We have
$$
\int e^{2x}\sin 3x dx = \int \sin 3x de^{2x}\frac{1}{2} \simeq \frac{1}{2}[ e^{2x}\sin 3x - 3 \int e^{2x} \cos 3x dx ]\\ = \frac{1}{2}e^{2x}\sin 3x - \frac{3}{2} \int e^{2x}\cos 3x dx;
$$
we have
$$
\int e^{2x}\cos 3x dx = \frac{1}{2}\int \cos 3x de^{2x} \simeq \frac{1}{2}e^{2x}\cos 3x + \frac{3}{2}\int e^{2x}\sin 3x dx;
$$
hence
$$
\frac{13}{4}\int e^{2x}\sin 3x dx \simeq \frac{1}{2}e^{2x}\sin 3x - \frac{3}{4}e^{2x}\cos 3x.
$$
