AP Calculus BC · Lesson 2 of 4
Integration techniques and improper integrals
Choose an integration method from the form of the integrand, and treat an infinite bound as a limit.
Choosing a method
Substitution is appropriate when you see a function and a constant multiple of its derivative. Integration by parts is appropriate for a product that becomes simpler when one factor is differentiated. Partial fractions rewrite a proper rational expression as a sum of simpler fractions after factoring the denominator.
After integrating, differentiate your result. An indefinite integral still needs a constant. A definite integral may change its limits when you substitute; forgetting the new limits is a frequent error.
Improper integrals
An integral to infinity is defined as a limit of integrals to a finite bound b, as b approaches infinity. You cannot substitute “infinity” as an ordinary number. The integral of 1/x² from 1 to infinity is the limit of 1 − 1/b, which is 1, so it converges.
An integral can also be improper at a finite endpoint where the integrand is unbounded. That case is likewise defined by a limit. If the limit is infinite or does not exist, the improper integral diverges.
Worked example
Evaluate the integral of 1/x² from 1 to infinity.
- Replace the upper bound by b: the integral from 1 to b is −1/b − (−1/1) = 1 − 1/b.
- Take the limit as b approaches infinity.
- The limit is 1, so the improper integral converges to 1.
Why this works. Write the limit explicitly. Infinity is not a number you plug into an antiderivative.
Check your understanding
- Name the integration method and the feature of the integrand that suggests it.
- Differentiate to verify an antiderivative.
- Rewrite an improper integral as a limit before evaluating it.