Mathematics.

integral calculus

Improper Integrals

Calculus II30 minDifficulty5 out of 10

You should know: integral, limit

Overview

An improper integral is a definite integral where either the interval of integration is infinite, or the integrand becomes unbounded somewhere in the interval. Such integrals are defined as a limit of proper (ordinary) integrals, and they may converge to a finite value or diverge.

Intuition

You can't directly plug ∞ into the Fundamental Theorem of Calculus, so instead you integrate up to a large but finite bound t, then ask what happens as t itself grows without bound. If the resulting expression settles down to a finite number, the improper integral converges to that limit; if it grows unboundedly or oscillates, it diverges.

Interactive Graph

1/x^2 — area under an infinite tail

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Formal Definition

Definition

Two types of improper integrals, both defined via limits:

∫a∞f(x) dx=lim⁡t→∞∫atf(x) dx\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty} \int_a^t f(x)\,dx
Type I (infinite interval)
∫abf(x) dx=lim⁡t→b−∫atf(x) dxif f is unbounded near b\int_a^b f(x)\,dx = \lim_{t\to b^-} \int_a^t f(x)\,dx \quad \text{if } f \text{ is unbounded near } b
Type II (unbounded integrand)

Applications

Escape velocity and gravitational potential energy calculations integrate force over an infinite distance, requiring improper integrals.

Worked Examples

  1. Rewrite as a limit of a proper integral.

    ∫1∞1x2dx=lim⁡t→∞∫1tx−2 dx\int_1^{\infty}\frac{1}{x^2}dx = \lim_{t\to\infty}\int_1^t x^{-2}\,dx

  2. Evaluate the antiderivative.

    =lim⁡t→∞[−1x]1t=lim⁡t→∞(−1t+1)= \lim_{t\to\infty}\Big[-\frac1x\Big]_1^t = \lim_{t\to\infty}\left(-\frac1t + 1\right)

  3. Take the limit as t→∞.

    =0+1=1= 0 + 1 = 1

Answer: The integral converges to 1.

Practice Problems

Difficulty 5/10

Determine whether ∫₁^∞ 1/x dx converges or diverges.

Difficulty 6/10

Evaluate ∫₀^∞ e^{-x} dx.

Common Mistakes

Common Mistake

Directly plugging in ∞ as if it were a number, e.g. writing [-1/x] from 1 to ∞ as -1/∞ - (-1/1) without taking a limit.

∞ is not a number that can be substituted; always express the improper integral as lim(t→∞) of a proper integral with upper bound t, then evaluate the limit.

Common Mistake

Missing that the integrand is unbounded somewhere strictly inside the interval (not just at an endpoint), such as 1/(x-2)² on [0,4].

Check the entire interval for singularities, not just the endpoints. If the integrand blows up at an interior point c, split the integral at c into two improper integrals, each of which must converge for the whole to converge.

Quiz

An improper integral over [a, ∞) is evaluated by:
An improper integral is said to CONVERGE when:

Summary

  • Improper integrals arise from infinite intervals of integration or unbounded integrands, and are defined as limits of proper integrals.
  • ∫ₐ^∞ f(x)dx = lim(t→∞) ∫ₐ^t f(x)dx; similarly for integrands unbounded near an endpoint.
  • The integral converges if the limit exists and is finite; otherwise it diverges.
  • p-integrals like ∫₁^∞ 1/xᵖ dx converge exactly when p > 1, a benchmark comparison test result.

References

  1. WebsiteWikipedia — Improper integral