← MATH 01131 · Calculus II
MATH 01131 · Advanced integration
Improper integrals
Definition
Integrals with infinite limits or unbounded integrands. Define as limits: ∫ₐ^∞ f = lim b→∞ ∫ₐᵇ f. Converges if the limit is finite; else diverges.
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Split at singularities: ∫₀¹ 1/√x needs lim a→0⁺ ∫ₐ¹.
Comparison / limit-comparison tests mirror series tests for tails.
p-integrals: ∫₁^∞ x⁻ᵖ converges iff p>1; ∫₀¹ x⁻ᵖ converges iff p<1.
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Attempt each problem first, then open the worked solution.
1.Decide convergence of ∫₁^∞ 1/x² dx and ∫₀¹ 1/x dx.