CalcRef

Techniques of Integration · Improper Integrals

p-Integral Test

∫1∞1xp dx converges if p>1, diverges if p≤1\int_1^{\infty} \frac{1}{x^p}\, dx \text{ converges if } p > 1 \text{, diverges if } p \leq 1

A fundamental convergence result: the improper integral of 1/xᵖ from 1 to ∞ converges if and only if p > 1.

  • 2 worked examples
  • 16 in Techniques of Integration

Worked examples

Does ∫₁^∞ 1/x^(3/2) dx converge?
  1. p = 3/2 > 1, so the integral converges.
  2. Value: [-2/√x]₁^∞ = 0-(-2) = 2

Answer: Converges; equals 2.

1 more worked examplePremium

Does ∫₁^∞ 1/√x dx converge?

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 16 Techniques of Integration formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the p-Integral Test?

A fundamental convergence result: the improper integral of 1/xᵖ from 1 to ∞ converges if and only if p > 1. It is one of the techniques of integration formulas in the CalcRef reference.

How do you use the p-Integral Test?

Worked example. Does ∫₁^∞ 1/x^(3/2) dx converge? p = 3/2 > 1, so the integral converges. Value: [-2/√x]₁^∞ = 0-(-2) = 2. Answer: Converges; equals 2.

Practice

Quiz yourself on Techniques of Integration

p-Integral Test is one of 16 Techniques of Integration formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 16 Techniques of Integration formulas