Math · Calculus II · Worked example

Evaluate an improper integral to infinity

Evaluate the integral of 1/x² from 1 to ∞, or show that it diverges.

\int_{1}^{\infty} \frac{d x}{x^{2}}

Write it as a limit

Replace ∞ by b and integrate over the ordinary interval [1, b].

lim_{b \to \infty} \int_{1}^{b} \frac{d x}{x^{2}}

Integrate on [1, b]

An antiderivative of 1/x² is −1/x, so the integral from 1 to b is −1/b − (−1) = 1 − 1/b.

Take the limit

As b → ∞, 1/b → 0, so the integral converges to 1.

Check with a large b

At b = 1000 the ordinary integral is already 0.999.

1 - \frac{1}{1000} = 0.999

Result

The integral converges: the integral of 1/x² from 1 to ∞ is 1.

Your turn

Evaluate the integral of e^(−2x) from 0 to ∞.

Show the answer and explanation

1/2.

The integral from 0 to b is 1/2 − e^(−2b)/2, and e^(−2b) → 0 as b → ∞.

Keep exploring

Limits & one-sided behavior opens with 1 − 1/b as b → ∞ and verifies the limit 1. Derivative & antiderivative checks confirms that −1/x is an antiderivative of 1/x² for x > 0.

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Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax Calculus Volume 2 — Improper integrals

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Check the limit in Limits & one-sided behavior Check the antiderivative Open worked example on a board Fundamental Theorem in Math Reference

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