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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

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.

### Lagrange (Prime) Notation

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:

- Change variables (substitute in
*u*for some function of*x*). - Apply an appropriate pattern to find the indefinite integral.
- Put the original variable back (substitute the function of
*x*back in for*u*).

The trickiest part is usually figuring out which function we want to replace with

*u*. Here are two guidelines that might help:*u*should be as complicated as possible, but still an "inside" function.*u*' should be similar to something else already in the function.

- Change variables (substitute in
### Leibniz (Fraction) Notation

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:

- Change variables (substitute in
*u*for some function of*x*). - Integrate.
- Put the original variable back (substitute the function of
*x*back in for*u*).

- Change variables (substitute in

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