Finding Antiderivatives
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.
Changing the Variable and Definite Integrals
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?
Special Transformations for
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