Study Guide

Indefinite Integrals - Integration by Substitution: Indefinite Integrals

Integration by Substitution: Indefinite Integrals

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.

  • 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.
  • 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).

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