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.

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.

Does ∫₁^∞ 1/√x dx converge?
  1. p = 1/2 ≤ 1, so the integral diverges.

Answer: Diverges.

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