Short on time? This page is the recap. For the graph, the definitions table and worked examples, use the full study guide. BC-only material. Prerequisites (limits, L’Hospital’s rule 4.7, FTC 6.7, techniques 6.9 to 6.12) are on the Calculus BC roadmap.
Recap
- An integral is improper if a limit of integration is ±∞, or if the integrand is unbounded at an endpoint or at a point inside the interval.
- Method: replace the problem point with a letter, find the definite integral, then take the limit.
- Converges: the limit is a finite number (the value). Diverges: the limit is infinite or does not exist.
- An infinitely long or infinitely tall region can still have a finite area.
Key relationships
| Case | Rewrite as | Note |
|---|---|---|
| Infinite upper limit | lim as b → ∞ of ∫ from a to b of f(x) dx | Same idea for −∞ below |
| Asymptote at the endpoint a | lim as t → a⁺ of ∫ from t to b of f(x) dx | One-sided limit from inside the interval |
| Asymptote at c inside [a, b] | ∫ from a to c + ∫ from c to b | Both parts must converge |
| From −∞ to ∞ | ∫ from −∞ to 0 + ∫ from 0 to ∞ | Not one symmetric limit |
| p-integral on [1, ∞) | ∫ 1/xᵖ dx = 1/(p − 1) if p > 1 | Diverges if p ≤ 1 |
| p-integral on (0, 1] | ∫ 1/xᵖ dx = 1/(1 − p) if p < 1 | Diverges if p ≥ 1 |
| Example | ∫ from 1 to ∞ of 4/(x(x + 2)) dx = 2 ln 3 | Combine logs before the limit |
Mistakes to avoid
- Plugging in ∞. Use a letter and limit notation.
- Missing a hidden asymptote. ∫ from −1 to 1 of 1/x² dx diverges; it is not −2.
- ∞ − ∞. Combine log terms into one log first.
- “f → 0 so it converges.” 1/x is the counter-example.
- Symmetric limits for ∫ from −∞ to ∞: split, and check both halves.
- Skipping L’Hospital’s rule for b e^(−b) or (ln b)/b.
Quick self-check
- Evaluate ∫ from 1 to ∞ of 3/x⁴ dx. (lim as b → ∞ of (1 − 1/b³) = 1)
- Evaluate ∫ from 0 to 4 of 1/√x dx. (lim as t → 0⁺ of (4 − 2√t) = 4)
- Evaluate ∫ from 0 to ∞ of e^(−x/5) dx. (lim as b → ∞ of (5 − 5e^(−b/5)) = 5)
- Does ∫ from 2 to ∞ of 1/(x − 1) dx converge? (No: ln(b − 1) → ∞, so it diverges)
Next: practice questions.