6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002

The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.

Example 1: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem!

Example 2: One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is . Note that this only worked because of the 2x in the original. Many integrals can not be done by substitution.

Example 3: Solve for dx.

Example 4:

Example 5: We solve for because we can find it in the integrand.

Example 8:

The technique is a little different for definite integrals. Example 10: The technique is a little different for definite integrals. new limit We can find new limits, and then we don’t have to substitute back. new limit We could have substituted back and used the original limits.

Wrong! The limits don’t match! Example 10: Using the original limits: Leave the limits out until you substitute back. Wrong! The limits don’t match! This is usually more work than finding new limits

Example 10: Don’t forget to use the new limits.

In another generation or so, we might be able to use the calculator to find all integrals. Until then, remember that half the AP exam and half the nation’s college professors do not allow calculators. You must practice finding integrals by hand until you are good at it! p