Math · Calculus II · Concept
Improper integrals and convergence
An integral is improper when an endpoint is infinite or the integrand blows up somewhere on the interval. Replace the troublesome endpoint by a variable, integrate over the ordinary interval that results, and take the limit. If the limit is a finite number the integral converges to it; otherwise it diverges. The p-integrals ∫ from 1 to ∞ of dx/xᵖ, which converge exactly when p > 1, are the benchmarks for comparison.
Infinite intervals
The integral from a to ∞ means the limit of the integral from a to b as b → ∞. The area under a curve over an infinite interval can be finite, as under 1/x², or infinite, as under 1/x.
Infinite integrands
If f blows up at an endpoint a, integrate from t to b and let t → a from inside the interval. If f blows up at an interior point c, split the integral at c; it converges only if both pieces do.
The p-integrals
The integral from 1 to ∞ of dx/xᵖ converges to 1/(p − 1) when p > 1 and diverges when p ≤ 1. Near 0 the pattern flips: the integral from 0 to 1 of dx/xᵖ converges to 1/(1 − p) when p < 1 and diverges when p ≥ 1.
Comparison
If 0 ≤ f(x) ≤ g(x) and the integral of g converges, so does the integral of f; if the integral of f diverges, so does the integral of g. Comparing with a p-integral or with e^(−x) often settles convergence without finding an antiderivative.
Improper integrals and series
The integral test links the integral from 1 to ∞ of f with the series f(1) + f(2) + ⋯: for positive decreasing f, both converge or both diverge. That is why the p-series ∑1/nᵖ behaves like the p-integral.
Common mistakes
- Applying the Fundamental Theorem across a vertical asymptote: the integral of 1/x² from −1 to 1 is not −2; it diverges.
- Writing F(∞) as if ∞ were a number instead of taking a limit.
- Deciding that an integral converges because the integrand tends to 0: the integral of 1/x from 1 to ∞ diverges.
- Adding a divergent piece to a convergent one after splitting at an asymptote, as if they could cancel.
Key terms
- Improper integral
- An integral over an infinite interval, or of a function that blows up, defined as a limit. If it has trouble on both sides of a point, each part must converge on its own; opposite infinities don’t cancel.
- Convergence
- Settling on a finite value in a limit. A sequence converges when its terms approach a number; a series converges when its partial sums do. Numbers that look stable are a hint, not a proof.
- Divergence
- Not approaching a finite value. A divergent sequence or series may grow without bound or keep oscillating, so divergence doesn’t always mean heading to +∞.
- p-series
- The series Σ 1/nᵖ. It converges when p > 1 and diverges when p ≤ 1, as the harmonic series (p = 1) shows.
- Comparison test
- A test for series of positive terms: a series smaller, term by term, than a convergent series converges, and one larger than a divergent series diverges. The limit comparison test compares two series through the limit of the ratio of their terms.
- Vertical asymptote
- A vertical line x = a that the graph approaches as the function’s values grow without bound near a, often where a denominator is zero. The graph is not joined across it.
Work through an example
Evaluate the integral of 1/x² from 1 to ∞, or show that it diverges.
Evaluate an improper integral to infinity →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Check the limit in Limits & one-sided behavior Check the antiderivative Open worked example on a board Fundamental Theorem in Math ReferenceYour existing work stays on this device. Examples open as editable copies.