Integration by Parts with Complex Exponentials

[abney317]

I'm looking at this problem here. (Exam practice, move to homework if you want...)

First part is easy, but it's the second part that I can't quite figure out.


I'm trying to get from ∫xe^2x cos(2x)dx to this answer:
¼e^2x(x cos(2x) + x sin(2x) - ½sin(2x))+C

 

[pwsnafu]

Do you know Euler's formula?

 

[HallsofIvy]

$$cos(2x)= \frac{e^{2ix}+ e^{-2ix}}{2}$$
so
$$\int xe^{2x} cos(2x)dx= \frac{1}{2}\int x(e^{2(1+ i)x}+ e^{2(1- i)x})dx$$