How can I integrate e^x sin(x)?

This integral is a particularly difficult integral that has shown up in STEP questions before. It's not hard because of how difficult the actual calculations are, but more because of how hard it is to spot the 'trick' to doing this integral.
The trick is integration by parts twice. You can integrate by parts by letting u=e^x and v'=sin(x). You end up with the integral of e^x cos(x), which you integrate with exactly the same method. Then, you end up with, if I is the integral of e^x sin(x),
I=e^x sin(x) - e^x cos(x) - I + c, which implies that I=e^x (sin(x)-cos(x)) + c.