Calculus BC · Unit 6: Integration and Accumulation of Change · 14 min read · Updated 2026-05-11

Improper Integrals (BC only) — AP Calculus BC

AP Calculus BC · Unit 6: Integration and Accumulation of Change · 14 min read

1. What is an Improper Integral? ★★☆☆☆ ⏱ 3 min

This topic is exclusive to AP Calculus BC and does not appear on the AP Calculus AB exam. It accounts for approximately 6-8% of the total AP Calculus BC exam score, and appears in both multiple-choice and free-response sections, often combined with other topics like integration techniques or infinite series.

2. Improper Integrals with Infinite Bounds (Type 1) ★★☆☆☆ ⏱ 4 min

Type 1 improper integrals have at least one infinite bound of integration, and the integrand is continuous on the entire interval. By definition:

If $f$ is continuous on $[a, \infty)$:

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

If $f$ is continuous on $(-\infty, b]$:

\int_{-\infty}^b f(x) dx = \lim_{a \to -\infty} \int_a^b f(x) dx

If $f$ is continuous on $(-\infty, \infty)$, split the integral at any finite point $c$:

\int_{-\infty}^\infty f(x) dx = \int_{-\infty}^c f(x) dx + \int_c^\infty f(x) dx

The original integral converges **only if both separate integrals converge**.

3. Improper Integrals with Discontinuous Integrands (Type 2) ★★★☆☆ ⏱ 4 min

Type 2 improper integrals have finite integration bounds, but the integrand has an infinite discontinuity (vertical asymptote) at one or more points in the interval. We use the same limit-based approach, but approach the discontinuity instead of moving a bound to infinity.

If $f$ is continuous on $(a, b]$ with an infinite discontinuity at $a$:

\int_a^b f(x) dx = \lim_{c \to a^+} \int_c^b f(x) dx

If $f$ is continuous on $[a, b)$ with an infinite discontinuity at $b$:

\int_a^b f(x) dx = \lim_{c \to b^-} \int_a^c f(x) dx

If the discontinuity is at an interior point $c \in (a,b)$, split the integral into two improper integrals at $c$; both must converge for the original integral to converge.

4. Convergence Testing: p-Test and Comparison Test ★★★☆☆ ⏱ 3 min

On many AP exam questions, you only need to determine if an improper integral converges or diverges, not calculate its exact value. Two key tools for this are the p-test for power functions and the comparison test for general positive integrands.

**p-test for infinite bounds ($a>0$):** $\int_a^\infty \frac{1}{x^p} dx$ converges if $p>1$, diverges if $p \leq 1$. The function must decay fast enough as $x$ grows to have finite area.
**p-test for discontinuity at 0 ($a>0$):** $\int_0^a \frac{1}{x^p} dx$ converges if $p<1$, diverges if $p \geq 1$. The function cannot blow up too fast near $x=0$ to have finite area.

The comparison test applies to positive functions: if $0 \leq f(x) \leq g(x)$ for all $x$ in the interval:

If $\int g(x) dx$ converges, then $\int f(x) dx$ also converges.
If $\int f(x) dx$ diverges, then $\int g(x) dx$ also diverges.

5. Mixed and Applied Improper Integral Problems ★★★★☆ ⏱ 4 min

📐 Worked Example

Let $f(x) = \frac{1}{(x-1)(x-3)}$. (a) Explain why $\int_0^4 f(x) dx$ is improper. (b) Split the integral into a sum of limits of proper integrals. (c) Determine if the integral converges or diverges.

(a) The integrand has infinite discontinuities (vertical asymptotes) at $x=1$ and $x=3$, both inside the integration interval $[0,4]$, so the integral is improper by definition.
(b) Split the integral at the two discontinuities, resulting in four separate limits:
$\int_0^4 f(x) dx = \lim_{a \to 1^-} \int_0^a f(x) dx + \lim_{b \to 1^+} \int_b^2 f(x) dx + \lim_{c \to 3^-} \int_2^c f(x) dx + \lim_{d \to 3^+} \int_d^4 f(x) dx$
(c) Use partial fraction decomposition to get $f(x) = -\frac{1}{2(x-1)} + \frac{1}{2(x-3)}$. Evaluate the first limit:
$\lim_{a \to 1^-} \int_0^a \left(-\frac{1}{2(x-1)} + \frac{1}{2(x-3)}\right) dx = \lim_{a \to 1^-} \left[ -\frac{1}{2}\ln|x-1| + \frac{1}{2}\ln|x-3| \right]_0^a$
As $a \to 1^-$, $\ln(1-a) \to -\infty$, so $-\frac{1}{2}\ln(1-a) \to +\infty$. The first limit diverges, so the entire integral diverges.

📐 Worked Example

The total work required to move a 1000 kg spacecraft from Earth's surface to infinitely far away is given by: $W = \int_R^\infty \frac{GMm}{r^2} dr$, where $G = 6.67 \times 10^{-11}$, $M = 5.97 \times 10^{24}$, $m = 1000$, $R = 6.37 \times 10^6$. Calculate the total work in joules.

Rewrite the improper integral as a limit by definition:
$W = GMm \lim_{b \to \infty} \int_R^b r^{-2} dr$
The antiderivative of $r^{-2}$ is $-r^{-1} = -\frac{1}{r}$. Evaluate the limit:
$\lim_{b \to \infty} \left(-\frac{1}{b} + \frac{1}{R}\right) = 0 + \frac{1}{R} = \frac{1}{R}$
Substitute the given values:
$W = \frac{GMm}{R} = \frac{(6.67 \times 10^{-11})(5.97 \times 10^{24})(1000)}{6.37 \times 10^6} \approx 6.25 \times 10^{10}$
The total work (escape energy) is approximately $6.25 \times 10^{10}$ joules.

Common Pitfalls

Why: Confuses the Cauchy principal value with the formal AP definition of convergence for two-sided infinite improper integrals.

Why: Mixes up the p-test conditions for infinite bounds vs. discontinuity at 0.

Why: Forgets to check for vertical asymptotes in the interior of the interval when bounds are finite.

Why: Misapplies the comparison test rules for convergence and divergence.

Why: Incorrectly assumes convergence of one part offsets divergence of another.

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