Improper integrals
Objective. Evaluate integrals with infinite bounds or singularities.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
An improper integral has an infinite limit of integration or an integrand that is undefined (tends to infinity) somewhere within the interval, and is evaluated as a limit of a proper integral approaching the problematic bound. For an infinite upper limit, ∫ₐ^∞ f(x)dx is defined as the limit as T→∞ of ∫ₐ^T f(x)dx, and the integral converges only if this limit exists finitely. For example, ∫₁^∞ 1/x² dx = lim(T→∞)[−1/x]₁^T = lim(T→∞)(−1/T+1) = 1, so this improper integral converges to 1.
Worked example
Evaluate the improper integral ∫₁^∞ 1/x³ dx, stating whether it converges.
- 1Write as a limit: lim(T→∞) ∫₁^T x⁻³ dx
- 2Integrate: ∫x⁻³dx = −x⁻²/2, so the definite integral is [−1/(2x²)]₁^T
- 3Evaluate: −1/(2T²) + 1/2
- 4As T→∞, −1/(2T²)→0, so the integral converges to 1/2
Answer: Converges to 1/2
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Practice
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Start practisingCommon mistake. Students substitute infinity directly into the antiderivative without taking a proper limit, which can mask divergence. Always write the limit explicitly as T→∞ (or similar) and evaluate it carefully before concluding convergence or divergence.
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