CalcRef

Techniques of Integration · Improper Integrals

Comparison Test for Integrals

0≤f(x)≤g(x):∫g conv.⇒∫f conv.;∫f div.⇒∫g div.0 \leq f(x) \leq g(x): \int g \text{ conv.} \Rightarrow \int f \text{ conv.}; \quad \int f \text{ div.} \Rightarrow \int g \text{ div.}

If 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f converges. If ∫f diverges, then ∫g diverges.

Worked examples

Does ∫₁^∞ e⁻ˣ/x dx converge?
  1. For x ≥ 1: e⁻ˣ/x ≤ e⁻ˣ
  2. ∫₁^∞ e⁻ˣ dx = 1/e (converges)
  3. By comparison, ∫₁^∞ e⁻ˣ/x dx converges.

Answer: Converges by comparison with ∫ e⁻ˣ dx.

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