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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 derivatives.

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.
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