TITLE: How does differentiation under the integral sign work?
QUESTION [5 upvotes]: From what I gather, it looks like you can use the method when your function depends on a variable and also a parameter. If you are given some definite integral that depended only on a variable, how would you determine whether the method would be useful or not and how would you determine where to insert a parameter?
Also, geometrically, how does differentiation under the integral sign help you evaluate the definite integral?

REPLY [5 votes]: This is a rather involved question, and in lieu of an original answer I refer you to the following excellent expository note: http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf.