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.
Once an antiderivative in terms of is found, we have a choice of methods. We can
(a) rewrite our antiderivative in terms of the original variable , and then evaluate the antiderivative at the integration endpoints and subtract, or
(b) change the integration endpoints to values of , and evaluate the antiderivative in terms of before subtracting.
If the original integral had endpoints and , and we make the substitution and , then the new integral will have endpoints and and
(original integrand) becomes (new integrand) .
To evaluate , we can put . Then so the integral becomes .
(a) Converting our antiderivative back to the variable and evaluating with the original endpoints:
(b) Converting the integration endpoints to values of : when , then , and when , then so
Both approaches typically involve about the same amount of work and calculation.
Practice 4: If the original integrals in Example 4 had endpoints (a) to , (b) to , and (c) and to , then the new integrals should have what endpoints?
The integrals of and occur relatively often, and we can find their antiderivatives with the help of two trigonometric identities for :
Solving the identies for and , we get and