Differentiation under the integral sign

[#1procrastinator]

http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf

If you have an integral that depends on one variable, how do you choose the parameter t and where to put it? (for example the example on the second page with the Euler factorial thing).

How do you know when this technique will work and when to use it? How common are the integrals with parameters in higher math?

Thanks