Calculus - Improper Integrals

Improper Integrals

Type 1: Infinite Intervals

$\int_a^{\infty} f(x)dx = \lim_{t \to \infty} \int_a^t f(x)dx$

$\int_{-\infty}^b f(x)dx = \lim_{t \to -\infty} \int_t^b f(x)dx$

$\int_{-\infty}^{\infty} f(x)dx = \int_{-\infty}^c f(x)dx + \int_c^{\infty} f(x)dx$

Type 2: Discontinuous Integrands

If f has discontinuity at a: $\int_a^b f(x)dx = \lim_{t \to a^+} \int_t^b f(x)dx$

Comparison Test

If 0 ≤ g(x) ≤ f(x) for x ≥ a:

• If ∫[a to ∞] f(x)dx converges, then ∫[a to ∞] g(x)dx converges
• If ∫[a to ∞] g(x)dx diverges, then ∫[a to ∞] f(x)dx diverges

p-Integral: ∫[1 to ∞] 1/x^p dx converges if p > 1, diverges if p ≤ 1.