Mathfolis

Improper Integrals

Unit 6 · Integration & Accumulation

Improper Integrals (BC Only)

An improper integral has an infinite limit of integration (Type 1) or an integrand discontinuous on the interval (Type 2). In both cases, replace the problematic bound with a variable and take a limit. If the limit is finite, the integral converges; otherwise it diverges.

Type 1 — Infinite Limits

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^\infty f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx
∫−∞bf(x) dx=lim⁡a→−∞∫abf(x) dx\int_{-\infty}^b f(x)\,dx = \lim_{a\to-\infty}\int_a^b f(x)\,dx

Type 2 — Discontinuous Integrand

Discontinuity at b:
∫abf(x) dx=lim⁡c→b−∫acf(x) dx\int_a^b f(x)\,dx = \lim_{c\to b^-}\int_a^c f(x)\,dx
Discontinuity at a:
∫abf(x) dx=lim⁡c→a+∫cbf(x) dx\int_a^b f(x)\,dx = \lim_{c\to a^+}\int_c^b f(x)\,dx
Caution: Never plug ∞ directly into the antiderivative — always write the limit explicitly first.
AP Tip: Useful benchmark: ∫₁^∞ 1/xᵖ dx converges iff p > 1.
Type 1

Integrals with Infinite Limits

Replace ∞ with a variable b, integrate on [a, b], then take the limit as b → ∞.

Example 1

Does the integral converge or diverge? If it converges, find its value.

∫1∞1x2 dx\int_1^\infty \frac{1}{x^2}\,dx
Example 2

Does the integral converge? If so, find its value.

∫0∞xe−x dx\int_0^\infty xe^{-x}\,dx

Practice more of this type— AI-generated · infinite problems

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Type 2

Discontinuous Integrand

Identify the discontinuity, replace that endpoint with a variable, integrate, then take the one-sided limit.

Example 3

Does the integral converge? If so, find its value.

∫011x dx\int_0^1 \frac{1}{\sqrt{x}}\,dx

Practice more of this type— AI-generated · infinite problems

Generate Problems →
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Improper Integrals | AP Calculus BC — Mathfolis