Evaluating Improper Integrals
Define improper integrals using separate finite one-sided limits, rather than cancellation across a pole.
College Board Topic 6.13 · CED p. 125 (PDF p. 130) · CED effective Fall 2020; project course outline 2026–27.
BC extension: an integral is improper when an endpoint is infinite or the integrand is unbounded somewhere in the interval. Replace the problem spot with a finite bound, integrate, then take a limit. It converges only when every required one-sided limit is finite. An infinite region can have finite area; the length of an interval alone does not decide.
For ∫₁^∞ x^(−p)dx, the limit converges exactly when p>1. For ∫₀¹ x^(−p)dx, the singular endpoint converges when p<1. An interior singularity must be split into two separate improper integrals. A symmetric cancellation is a principal value, not ordinary convergence: ∫₋₁¹ 1/x dx diverges because its two pieces diverge. Never substitute ∞ as though it were a real number, and never let a numerical integration key hide a pole.
Let's work through it
- Write ∫₁^∞ x⁻²dx as limᵦ→∞ ∫₁ᵇ x⁻²dx. Reason: Infinity is handled with a limit of finite intervals.
- Evaluate ∫₁ᵇ x⁻²dx=1−1/b. Reason: The antiderivative is −1/x.
- The limit is 1, so the integral converges. Reason: The defining limit exists and is finite.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Does ∫₁^∞ 1/x dx converge?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Does ∫₁^∞ 1/x dx converge?
Worked answer and reason
No; ln b approaches infinity.
The limiting definite-integral values are unbounded.
Retrieval 2
Can opposite infinities cancel across an interior singularity?
Worked answer and reason
No.
Each one-sided improper integral must converge separately.
Catch the tempting mistake
- An antiderivative evaluated at the endpoints always settles an improper integral.
- The Fundamental Theorem on a continuous finite interval must be followed by the appropriate limits. A singularity inside the interval requires splitting first.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 26 · No calculator · analytic
- Unit 6 problem 27 · No calculator · analytic
- Unit 6 problem 28 · No calculator · analytic
- Unit 6 problem 29 · No calculator · verbal
- Unit 6 problem 30 · No calculator · analytic
For the teacher
- Use both an infinite endpoint and a finite singular endpoint. Contrast convergence with symmetric principal-value cancellation. Have students locate singularities before any calculator entry.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
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Integrand
The integrand is the function or expression being integrated. The differential identifies the variable of integration. For example, in the integral of x² with respect to x, the integrand is x². Bounds belong to the integral’s interval rather than to the integrand itself.