Concept explainerCalculus I · Unit 3AIntermediate

Improper integrals are limits, not unusual notation

Replace infinite bounds or unbounded integrands with limits before evaluating, then decide whether the result converges.

LaTeX article Updated July 11, 2026

\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx

Two sources of impropriety

An interval may extend to infinity, or the integrand may become unbounded at an endpoint or inside the interval. Each issue must be replaced by a one-sided limit.

If a singularity lies inside the interval, split the integral there and require both pieces to converge independently.

Antiderivatives do not settle convergence

The Fundamental Theorem applies first on finite intervals where the integrand behaves appropriately. Only after evaluation do you take the defining limit.

An expression such as infinity minus infinity is not a cancellation; the pieces must be analyzed separately.

Comparison can avoid hard antiderivatives

For nonnegative functions, comparison with a known p-integral or another benchmark can prove convergence or divergence without an exact formula.

Worked example

Common mistakes

Plugging infinity into an antiderivative as if it were a number.
Failing to split at an interior singularity.
Declaring convergence because the integrand approaches zero.

Keep these ideas

Rewrite first as one or more limits.
Every improper piece must converge.
Comparison is often more useful than exact integration.