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University Math Lab: Systems & Structures FREE
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Improper Integrals
Learning Objectives
- Evaluate integrals with infinite limits
- Handle discontinuities
- Test convergence
Key Concepts
Improper Integral
Integral with infinite limit or discontinuity in integrand
Convergent
Limit exists and is finite
Theory
**Type 1 — Infinite limits:**
∫[a,∞) f(x)dx = lim(b→∞) ∫[a,b] f(x)dx
**Type 2 — Discontinuity at c in [a,b]:**
∫[a,b] f(x)dx = lim(t→c⁻) ∫[a,t] f(x)dx + lim(t→c⁺) ∫[t,b] f(x)dx
**p-test:** ∫[1,∞) 1/xᵖ dx converges iff p > 1
**Comparison test:** If 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f converges.
Examples
∫[1,∞) 1/x² dx
Solution: lim(b→∞) [-1/x]₁ᵇ = lim(b→∞) (-1/b + 1) = 1. Converges.