Find the indefinite integral:

Question

I have to find the indefinite integral of (e^x)(cos x). I tried using integration by parts but I keep getting the same integral over and over again.

Answer 1

It is correct that you get the integral again after two steps of integration by parts: $$ \int e^x \cos x dx = e^x \cos x + \int e^x \sin x dx =e^x \cos x + e^x \sin x - \int e^x \cos x dx $$ But note the opposite sign! Now move the integral to the LHS and get $$ \int e^x \cos x dx = \frac{1}{2}\left( e^x \cos x + e^x \sin x\right) $$