--- title: "Improper Integrals and Their Convergence" description: "Give meaning to an integral with an infinite limit or an unbounded integrand by evaluating a proper integral and taking a limit, then decide whether the result is finite." canonical: https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z source: https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Improper Integrals and Their Convergence Give meaning to an integral with an infinite limit or an unbounded integrand by evaluating a proper integral and taking a limit, then decide whether the result is finite. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z ## Ready when they can - Rewrite infinite-range integrals with separate limits where needed, evaluate them and find a cutoff from a tail error - Use the convergence threshold and finite-value formula for 1/x^p on [1, infinity), including finding p from a given value - Handle endpoint and interior singularities with the required separate one-sided limits ## Lesson: Improper integrals and finite limits Use your antiderivatives, including partial fractions, with finite cutoffs. An improper integral has an infinite bound or an integrand unbounded near an endpoint or interior point. Convergence requires every cutoff limit to be finite. For f continuous on [A, infinity), use lim(t -> infinity) ∫A^t f(x) dx. For f continuous on (-infinity,B], use lim(a -> -infinity) ∫a^B f(x) dx. Evaluate each proper integral before taking its limit. **Example.** For ∫2^infinity 4/x^2 dx, an antiderivative is -4/x. Integrating from 2 to t gives 2-4/t, whose limit is 2. For comparison, ∫2^t 1/x dx = ln(t)-ln(2), which grows without bound. The first improper integral converges; the second diverges. An integrand tending to 0 does not by itself establish convergence. *(drawing: For 4/x^2 from 2 to t, the formula 2-4/t gives both the finite values and their limit.)* For real p, ∫1^infinity x^(-p) dx converges exactly when p > 1. For p != 1, the integral to t is (t^(1-p)-1)/(1-p). If p > 1, t^(1-p) tends to 0, leaving 1/(p-1). If p < 1, the expression grows without bound. At p = 1, it is ln(t), which also grows without bound. To find p from a finite value V, solve 1/(p-1)=V. For c > 0 and t > 0, the omitted tail ∫t^infinity c/x^2 dx equals c/t. Using -c/x as an antiderivative, the integral from t to b is c/t-c/b; let b grow to obtain c/t. When approximating ∫A^infinity by ∫A^t, with 0 < A < t, this tail is the error. For an exact error E > 0, solve c/t=E to get t=c/E. Use one-sided cutoffs at singular endpoints. For ∫0^B 1/sqrt(x) dx with B > 0, integrate from a > 0: 2sqrt(B)-2sqrt(a). Let a -> 0 from above to get 2sqrt(B). At the right endpoint of ∫0^L 1/sqrt(L-x) dx, take b < L. The antiderivative -2sqrt(L-x) gives 2sqrt(L)-2sqrt(L-b). Let b -> L from below to get 2sqrt(L), for L > 0. If f is unbounded near c inside [A,B] and continuous elsewhere, split at c. Evaluate lim(u -> c from below) ∫A^u f(x) dx and lim(v -> c from above) ∫v^B f(x) dx independently. Add their values only if both are finite. Opposite infinities cannot cancel to define a convergent improper integral. **Example.** For 1/sqrt(|x-c|), the left antiderivative is -2sqrt(c-x), and the right is 2sqrt(x-c). On [A,B], with A < c < B, the separate limits are 2sqrt(c-A) and 2sqrt(B-c). Both are finite. For example, on [-9,16] with c=0, the limits are 6 and 8, giving 14. For f continuous on the real line, split ∫-infinity^infinity f(x) dx at 0. Take the left limit a -> -infinity and the right limit b -> infinity independently. Add only if both integrals converge. Symmetric cutoffs alone do not decide convergence: ∫-t^t x dx = 0, but the left integral is -t^2/2 and the right is t^2/2. Each is unbounded as t grows, so the improper integral diverges. **Recap.** Use finite cutoffs at infinite bounds and singularities. Evaluate each required limit separately. Add the values only when every required limit is finite. *Lesson and questions written for this dataset. Calculus texts and problem records were consulted for mathematical facts and question design.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Partial Fractions for Rational Integrands](https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c) ## Optional connections - [L'Hopital's Rule and Comparing Growth Rates](https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE) ## Opens up - [Monotone Convergence, Fatou and Dominated Convergence](https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe) - [The Laplace Transform](https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0) - [Convergence Tests for Series of Positive Terms](https://lightmysky.com/learn/mathematics/convergence-tests-for-series-of-positive-terms-mt_8N49Q4BlO8) - [Expectation and Variance by Integration](https://lightmysky.com/learn/mathematics/expectation-and-variance-by-integration-mt_kRSaKgp3bs) - [Evaluating Real Integrals by Residues](https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh) - [Volumes of Revolution: Disks, Washers and Shells](https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4) - [The Distribution of Primes and What Is Still Open](https://lightmysky.com/learn/mathematics/the-distribution-of-primes-and-what-is-still-open-mt_PJwm9kuQWQ)