Techniques of Integration · Improper Integrals

Comparison Test for Integrals

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.

2 worked examples
16 in Techniques of Integration

Worked examples

Does ∫₁^∞ e⁻ˣ/x dx converge?

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

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

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Does ∫₁^∞ (2 + sin x)/x dx converge?

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Common questions

What is the Comparison Test for Integrals?

If 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f converges. If ∫f diverges, then ∫g diverges. It is one of the techniques of integration formulas in the CalcRef reference.

How do you use the Comparison Test for Integrals?

Worked example. Does ∫₁^∞ e⁻ˣ/x dx converge? For x ≥ 1: e⁻ˣ/x ≤ e⁻ˣ. ∫₁^∞ e⁻ˣ dx = 1/e (converges). By comparison, ∫₁^∞ e⁻ˣ/x dx converges. Answer: Converges by comparison with ∫ e⁻ˣ dx.

Practice

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