CalcRef

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.

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)

Evaluate ∫₁^∞ 1/x dx.
  1. lim(t→∞) [ln x]₁ᵗ = lim(t→∞) ln t = ∞

Answer: Diverges

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