Calculus · step 10 of 13

Improper integrals & convergence

An integral is improper when a limit is infinite or the integrand blows up inside the range. The definition is always a limit: ∫₁^∞ f = lim_{b→∞} ∫₁^b f. Convergence depends delicately on the exponent — ∫₁^∞ x^(−p) dx converges exactly when p > 1, while ∫₀¹ x^(−p) dx converges exactly when p < 1, the opposite condition.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identify the impropernessInfinite limit, or a vertical asymptote. If the blow-up is interior, split the integral at that point.
Replace the bad endpoint by a variableWrite ∫₁^b, evaluate normally, then take the limit as b → ∞ (or as b approaches the singularity).
Evaluate the limitA finite limit means convergence and gives the value; an infinite one means divergence.
Use comparison when the antiderivative is hardIf 0 ≤ f ≤ g and ∫g converges, so does ∫f. Compare against the standard power integrals.

Worked example

Evaluate ∫₁^∞ 1/x² dx.

Rewrite as lim_{b→∞} ∫₁^b x⁻² dx.
Antiderivative: −1/x.
Evaluate: [−1/x]₁^b = −1/b + 1.
As b → ∞, −1/b → 0.

Answer. The integral converges to 1. By contrast ∫₁^∞ 1/x dx diverges, since ln b → ∞.

Where marks get dropped

These are the specific errors that cost credit on improper integrals & convergence questions — QED's rubric penalises each of them separately.

Applying the fundamental theorem across an interior singularity. ∫₋₁¹ 1/x² dx is not 0 + something — it must be split at 0 and both pieces diverge.
Forgetting to write the limit. Substituting ∞ directly is not valid notation and loses method marks.
Mixing up the two power conditions. Convergence at infinity needs p > 1; convergence at a singularity at 0 needs p < 1.

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Improper integrals & convergence — frequently asked questions

Why does ∫₁^∞ 1/x diverge but ∫₁^∞ 1/x² converge?

Because ln b grows without bound while 1/b tends to 0. The extra power makes the tail shrink fast enough to have finite total area.

What if both endpoints are improper?

Split at any convenient interior point and require BOTH pieces to converge. If either diverges, the whole integral does.

What is a Cauchy principal value?

A symmetric limit around a singularity that can be finite even when the integral diverges. It is a different object and must be labelled as such.

The rest of Calculus

Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.

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