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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?

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I don't think there is a better answer to the question of how to know if the method of diff. under the integral sign would be useful other than "experience". After each example is read, ask yourself "why" it worked. – KCd Feb 16 '13 at 5:10

This is a rather involved question, and in lieu of an original answer I refer you to the following excellent expository note:

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