AP Calculus BC glossary
Improper integral
An improper integral is one with an infinite bound of integration or an integrand that grows without bound somewhere on the interval. It is defined as a limit of ordinary definite integrals, and it converges if that limit is finite and diverges otherwise.
The procedure is always the same: replace the problematic bound with a variable, evaluate the resulting definite integral, then take the limit. Skipping the limit notation costs credit even when the arithmetic is right.
If the integrand is unbounded at an interior point, split the integral there and treat each piece separately. Both pieces must converge for the whole integral to converge.
The mistake
Substituting infinity directly as if it were a number. Infinity is not a value you can plug in, and the limit is the entire content of the definition.
Appears in: Unit 6: Integration and Accumulation