What trick is used to integrate this?

[flyingpig]

Homework Statement

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

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?

 

[SammyS]

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

 

[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.