Solution to the cool integration by parts problem

This problem requires a bit more ingenuity.

Start out using integration by parts as you normally would.

You may notice that the new integral acquired still poses the same issue we started out (but with cosine instead of sine). Let us do integration by parts once more and see what happens

Now I need to substitute this back in. I need to also make sure to distribute the negative.

Now we have the exact same integral we started out with on both side. all we really need to do is add the integral of e^(x)sin(x) to both sides of the equation. Then we'll have 2 times the integral of e^(x)sin(x) on one side. Simply divide by two and the whole thing will be solved.

Is your mind blown or what?