Question

How do you find the antiderivative of `(cos 2x)e^(cos 2x)`?

Answer 1

I don't think you can (in finite form).

You can find the antiderivative of `(sin2x)e^cos(2x)` or of `(cos2x)e^sin(2x)` by substitution.

The antiderivative of `(sin2x)e^cos(2x)` is #-1/2e^cos(2x)+C #.
(Let `u=cos(2x)` , so `du=-2sin(2x)dx`)

The antiderivative of `(cos2x)e^sin(2x)` is #1/2e^sin(2x)+C #.
(Let `u=sin(2x)`, so `du=2cos(2x)dx`)