Calculus I · Unit 3A · lesson

Improper Integrals with Unbounded Integrands

Concept

Learning objectives

Split at singularities, use one-sided limits, and require every piece to converge.

Improper Integrals with Unbounded Integrands

Explanation

An infinite spike may or may not have finite accumulation

When an integrand becomes unbounded at an endpoint or inside the interval, the ordinary definite-integral notation hides a limit. Replace the problematic boundary by a nearby finite value and approach it from the correct side. If the singularity lies inside the interval, split the integral there and require both one-sided improper integrals to converge.

A graph can look alarming without determining the answer. A very tall but sufficiently narrow spike may have finite area, while a milder-looking singularity may diverge. The decision comes from the limiting calculation. Never cancel divergences from opposite sides unless a different concept, such as a principal value, has been explicitly introduced; ordinary improper-integral convergence requires each piece to exist independently.

If $f$ is unbounded at $c\in[a,b]$ , define

\int_a^bf(x)\,dx =\lim_{t\to c^-}\int_a^tf(x)\,dx +\lim_{s\to c^+}\int_s^bf(x)\,dx,

provided both one-sided integrals converge.

Worked example

Integrable vertical blow-up

\int_0^1\frac1{\sqrt{x}}\,dx =\lim_{a\to0^+}[2\sqrt{x}]_a^1=2.

The function becomes unbounded at $0$ , but the area remains finite.

Common mistake

Never evaluate across a singularity with a single antiderivative subtraction. Split the interval first. Cancellation between two divergent sides does not make the ordinary improper integral converge.

Interactive checku3a-improper-sing-01

Does $\int_0^1x^{-1/2}\,dx$ converge?

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Show hint

The exponent corresponds to a p-integral near zero with $p=1/2<1$ .

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After the explanation

Use the section idea

Reading lens

Make approximation error and limiting behavior explicit rather than hiding them behind a calculator result or an infinity symbol.

Mental model

Numerical rules replace a curve with simple local shapes; improper integrals replace a forbidden endpoint or infinite interval with a limit.

Decision

Choose the rule and partition, estimate scale and sign, or write the correct defining limit before evaluating.

Common trap

Treating an approximation as exact, using Simpson's Rule with an invalid partition, or substituting infinity as though it were a number.

Check yourself

Can you defend the estimate's scale or the improper integral's convergence from a written calculation?

Source & rights

Original instruction with traceable references.

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