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Integration by substitution is a way of undoing the chain rule. This is a once-in-a-lifetime opportunity to learn derivatives inside and out, forwards and backwards. Exciting, eh? Learning integration by substitution is almost as exciting as winning Deal or No Deal. Given a derivative that was produced by the chain rule, integration by substitution lets us work backwards to find an antiderivative. This method is also called u-substitution because we usually use u instead of some other letter.
We've done some integration by substitution already. We call it thinking backwards.
We can do integration by substitution using either Lagrange notation (primes u' ) or Leibniz notation (fractions like ). While your teacher might have a preference, we don't care which of these ways you use. Just be careful not to mix them up. You can use primes (u' ) or you can use fractions , but don't use both in the same problem.
When we use the chain rule to take derivatives, there are some patterns that show up a lot. Some examples are
We can use these patterns to find indefinite integrals.
The general strategy for integration by substitution has three steps:
The trickiest part is usually figuring out which function we want to replace with u. Here are two guidelines that might help:
To do integration by substitution using Leibniz notation, we think of the derivative function as a fraction of infinitesimally small quantities du and dx. We change variables by manipulating these infinitesimal quantities.
The general strategy is pretty much the same as before: