6.13 Evaluating Improper Integrals

What makes an integral improper?

An improper integral has an infinite limit of integration (like ∫₁^∞) or an integrand that's unbounded (has a vertical asymptote) somewhere on the interval. Since Riemann sums can't handle infinity or infinite discontinuities directly, we evaluate these using limits.

Evaluating with limits

For ∫ₐ^∞ f(x)dx, replace ∞ with a variable b and take the limit as b→∞: lim(b→∞) ∫ₐ^b f(x)dx. If the limit exists (is finite), the integral converges; if it doesn't exist or is infinite, it diverges. Same idea applies at a discontinuity — approach it with a limit, not by plugging in directly.

Discontinuities within the interval

If f has a vertical asymptote at c between a and b, split the integral at c into two improper integrals: ∫ₐᶜ + ∫ᶜᵇ, each evaluated with its own limit. The original integral converges only if both pieces converge; if either diverges, the whole integral diverges.

Key terms

Improper integral

A definite integral with an infinite limit of integration or an unbounded integrand (vertical asymptote) within the interval.

How to evaluate ∫ₐ^∞ f(x)dx

Rewrite as lim(b→∞) ∫ₐ^b f(x)dx and evaluate the limit.

Convergence

An improper integral converges if its corresponding limit exists and equals a finite value.

Divergence

An improper integral diverges if the limit is infinite or does not exist.

Handling interior discontinuities

Split the integral at the point of discontinuity into two improper integrals; both must converge for the original to converge.

Try an AP-style question

∫₁^∞ 1/x² dx = ?

1. 2
2. 1/2
3. Diverges
4. 1