Sofia makes and sells stuffed toys. She uses special fabric to cover a foam
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Computing an integral by substitution is the reverse of the chain rule for computing the derivative. Substitution is intended to rewrite a complicated-looking integral involving the derivative of some component expression as another much simpler integral. For example, if , then
and
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Integration by parts is the reverse of the product rule for derivatives:
Integrating both sides with respect to gives
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Personally, I think the best way to grasp the idea behind the two methods is to practice. You start to notice patterns to the point where knowing which is the "right" method to use becomes second nature.