Improper Integrals

Numerical Analysis · Axiom Academy

Handling infinite limits and singular integrands 1. Types of Improper Integrals Type 1: Infinite limits — ∫₀^∞ e^(-x) dx Type 2: Unbounded integrand — ∫₀¹ 1/√x dx Type 3: Both — ∫₀^∞ 1/(x√x) dx 2. Truncation for Infinite Limits Replace ∫ₐ^∞ f(x)dx with ∫ₐ^T f(x)dx for large T. Estimate the tail: Choose T so the tail contribution is below your tolerance. For singularities at endpoints, avoid evaluating f there: Use open quadrature formulas (midpoint rule) Or shift slightly away from the singularity Transform to remove the singularity or make the interval finite: If you know the singular behavior, subtract it off: Integrate the singularity analytically, the remainder numerically.

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