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Techniques of Integration · Improper Integrals

Improper Integral: Type I (Infinite Limit)

∫a∞f(x) dx=lim⁡t→∞∫atf(x) dx\int_a^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_a^t f(x)\, dx

When the upper (or lower) limit is infinite, replace it with a variable t and take the limit. If the limit is finite, the integral converges.

  • 2 worked examples
  • 16 in Techniques of Integration

Worked examples

Evaluate ∫₁^∞ 1/x² dx.
  1. lim(t→∞) ∫₁ᵗ x⁻² dx = lim(t→∞) [-1/x]₁ᵗ = lim(t→∞) (-1/t + 1) = 1

Answer: 1 (converges)

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Evaluate ∫₁^∞ 1/x dx.

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Common questions

What is the Improper Integral: Type I (Infinite Limit)?

When the upper (or lower) limit is infinite, replace it with a variable t and take the limit. If the limit is finite, the integral converges. It is one of the techniques of integration formulas in the CalcRef reference.

How do you use the Improper Integral: Type I (Infinite Limit)?

Worked example. Evaluate ∫₁^∞ 1/x² dx. lim(t→∞) ∫₁ᵗ x⁻² dx = lim(t→∞) [-1/x]₁ᵗ = lim(t→∞) (-1/t + 1) = 1. Answer: 1 (converges).

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