TITLE: Find $\int e^{-x}\cos x\,dx$ without using complex numbers
QUESTION [6 upvotes]: $\int e^{-x} \cos{x} dx $ - i know how to solve with Euler complex representation, but can't figure out how to solve with integration by parts or something.

REPLY [2 votes]: So, you are in a differential equations course?  Then: How about solving the differential equation $y' = e^{-x}\cos x$ using the method of undetermined coefficients, explained in that course?