Mr. Integral
Math notes/Calculus II/Improper and Numerical Integration

Improper Integrals and Convergence — Cheat sheet

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Core rule

Tail ∫1∞x−pdx\int_1^\infty x^{-p}dx: p>1p>1. Near zero ∫01x−pdx\int_0^1x^{-p}dx: p<1p<1.

Watch for

Split every interior singularity. Both pieces must converge. Symmetric cancellation defines a principal value, not ordinary convergence.