Question

Evaluate the integral e^x sin4x dx.

I know the answer is 1/17 e^x sin4x - 4/17 e^x cos4x + C but I don't know how to solve it out.

Answer 1

Use integration by parts:

I=∫e^xsin(4x)dx
=e^xsin(4x)-4∫e^xcos(4x)dx
=e^xsin(4x)-4e^xcos(4x)-16∫e^xsin(4x)dx
=e^xsin(4x)-4e^xcos(4x)-16I
Solve for I to get the above results.