What trick is used to integrate this?

1. Jul 20, 2011

flyingpig

1. The problem statement, all variables and given/known data

$$\int e^x \frac{1 + sin(x)}{1 + cos(x)}\;dx$$

3. The attempt at a solution

Apparently there is a trick involving e^x integrals like this

$$\int e^x[f(x) + f'(x)] dx = e^x f(x) + C$$

Now my question is not how to compute the integral above but where did this $$\int e^x[f(x) + f'(x)] dx = e^x f(x) + C$$ identity came from?? How does one know this exists?

2. Jul 20, 2011

SammyS

Staff Emeritus
Take the derivative of the result. That may help you see how this works.

3. Jul 20, 2011

JKreutz

Yeah, you get this result with integration "by parts", which is the name for using the product rule to rewrite integrals (hopefully in a form you can recognize and solve). If you take a second-semester single-variable calculus class you'll learn about this.