Find an antiderivative to the function f(x) = e^x cos(x)

We see a product of two things, so we should consider integration by parts (IBP). IBP is usually useful when either 1) you can differentiate or integrate one of the two things repeatedly to eventually get zero, such as x^n, or 2) one of the things can be differentiated or integrated to eventually give itself again, such as sin x. So here we can use IBP. I like the tabular method, as it ismuch faster and more effortless than most other setups that students use. This is much easier to explain during the interview rather than in this textbook, but essentially you set up a two column table of derivatives and integrals and keep differentiating and integrating until eventually you either see a zero or the product you started with, then add up all the terms.
The answer is any antiderivative of the form F(x) = e^x (cos x + sin x) / 2 + C, where C is a real number.