Read this section to see how you can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1-4.
Practice 2: We know , so it is reasonable to try .
, a result 7 times the function we want, so
divide the original "guess" by 7 and try it. .
so let's multiply our original "guess" by 2/3:
Practice 3:
Practice 4:
(a) so when . When , . (This integral is now very easy to evaluate. Why?)