Improper integrals appear when an interval is infinite or when the function becomes unbounded at a point in the interval. They matter because they decide whether a total accumulated quantity, such as area, probability, charge, or work, is finite or infinite. Instead of evaluating the integral in the usual way, we rewrite it as a limit and test whether that limit exists as a finite number.
Convergence means the infinite process produces a finite result, while divergence means it does not.
Key Facts
- Infinite interval definition: ∫_a^∞ f(x) dx = lim_(b→∞) ∫_a^b f(x) dx
- Vertical asymptote definition: ∫_a^b f(x) dx with f unbounded at c equals ∫_a^c f(x) dx + ∫_c^b f(x) dx, and both parts must converge.
- p-integral on [1,∞): ∫_1^∞ 1/x^p dx converges if p > 1 and diverges if p ≤ 1.
- p-integral near 0: ∫_0^1 1/x^p dx converges if p < 1 and diverges if p ≥ 1.
- Direct comparison: If 0 ≤ f(x) ≤ g(x) for large x and ∫ g(x) dx converges, then ∫ f(x) dx converges.
- Limit comparison: If f(x), g(x) ≥ 0 and lim_(x→∞) f(x)/g(x) = L with 0 < L < ∞, then ∫ f(x) dx and ∫ g(x) dx have the same behavior.
Vocabulary
- Improper integral
- An integral with an infinite interval of integration or an integrand that becomes unbounded on the interval.
- Convergence
- The property that an improper integral has a finite value when written as a limit.
- Divergence
- The property that an improper integral does not approach a finite value.
- Comparison test
- A test that proves convergence or divergence by bounding one nonnegative function above or below another.
- Limit comparison test
- A test that compares the long-term size of two nonnegative functions using the limit of their ratio.
Common Mistakes to Avoid
- Forgetting to rewrite the integral as a limit. An improper integral is not evaluated directly over infinity or through an asymptote.
- Using the p-integral rule with the wrong interval. For ∫_1^∞ 1/x^p dx convergence requires p > 1, but for ∫_0^1 1/x^p dx convergence requires p < 1.
- Applying direct comparison in the wrong direction. To prove convergence, the function must be smaller than a known convergent function; to prove divergence, it must be larger than a known divergent function.
- Ignoring a discontinuity inside the interval. If an integrand is unbounded at an interior point, the integral must be split there and every resulting improper integral must converge.
Practice Questions
- 1 Determine whether ∫_1^∞ 5/x^3 dx converges or diverges, and find its value if it converges.
- 2 Use limit comparison to decide whether ∫_2^∞ (3x + 1)/(x^3 - 4) dx converges or diverges.
- 3 Explain why ∫_1^∞ 1/(x + sin x) dx diverges by comparing its long-term behavior to a simpler improper integral.