Question

Evaluate the integral using method of integration by parts:

(integral sign)(e^(2x))sin(5x)dx

Answer 1

You can do the partial integration by integrating the e^(2x) term. Then you end up with an integral proportional to e^(2x)cos(5x). You then integrate that again using partial integration, again integrating the e^(2x) term. You then get the integral you started out with back plus some other terms. So, if we call the integral I, you have a relation of the form:

I = some terms - a I

where a is some constant. Then you solve for I:

I = (some terms)/(1+a)

Answer 2

If you are familiar with complex numbers, then you can write:

sin(5x) = Im[exp(5 i x)

The integral is thus the imaginary part of the integral of exp[(2 + 5 i)x]. This is:

Im {exp[(2 + 5 i)x]/(2 + 5 i)} =

1/29 exp(2x) [2 sin(5 x) -5 cos(5x)]