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.