A-level Further Maths Year 13 · Topic 2
Further calculus
Further calculus deals with harder integrals and infinite processes.
We will meet improper integrals, the mean value of a function and integrals giving inverse trigonometric functions.
Then we use Maclaurin series, limits and integration by reduction formulae.
- 5 subtopics
- 223 questions
- Free: 2.1 notes and 10 questions
In this topic
- 2.1Improper integralsFree below · 53 questions
- 2.2The mean value of a functionOpen in Full Marks · 51 questions
- 2.3Integrals giving inverse trigonometric and hyperbolic functions, and partial fractionsOpen in Full Marks · 15 questions
- 2.4Maclaurin series and series expansionsOpen in Full Marks · 51 questions
- 2.5Limits and reduction formulaeOpen in Full Marks · 53 questions
2.1 · Free notes
Improper integrals
Key points
- An integral is improper if a limit is infinite, or the integrand is undefined somewhere in the interval.
- Evaluate it as a limit: ∫₁^∞ f(x) dx = lim_{t→∞} ∫₁ᵗ f(x) dx. If the limit exists the integral converges; otherwise it diverges.
- Examples: ∫₁^∞ 1/x² dx = lim (1 − 1/t) = 1 (converges); ∫₁^∞ 1/x dx = lim ln t (diverges).
- Useful limits: as x → ∞, xᵏe^(−x) → 0 and (ln x)/x → 0; as x → 0⁺, x ln x → 0.
In more detail
- Worked example: ∫₀^∞ x e^(−x) dx: by parts, [−x e^(−x) − e^(−x)]₀ᵗ = 1 − (t + 1)e^(−t) → 1.
- Comparison: ∫₁^∞ 1/xᵖ dx converges if p > 1 and diverges if p ≤ 1.
- Exam tip: write the limit explicitly ("lim as t → ∞") and state that the integral converges or diverges.
Practice
Improper integrals: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. Why is ∫₁^∞ 1/x² dx improper?
- It is negative
- It has no antiderivative
- It equals zero
- Its upper limit is infinite
Show answer
D: Its upper limit is infinite. Infinite range.
2. Evaluate ∫₁^∞ 1/x² dx.
- ∞
- 1
- 0
- 2
Show answer
B: 1. lim (1 − 1/t) = 1.
3. Does ∫₁^∞ 1/x dx converge?
- No, it diverges
- Yes, to 1
- Yes, to 0
- Yes, to e
Show answer
A: No, it diverges. ln t → ∞.
4. Why is ∫₀¹ 1/√x dx improper?
- The upper limit is infinite
- It is negative
- The integrand is undefined at x = 0
- It is a polynomial
Show answer
C: The integrand is undefined at x = 0. 1/√0 is undefined.
5. Evaluate ∫₀¹ 1/√x dx.
- ∞
- 1
- 1/2
- 2
Show answer
D: 2. [2√x] from 0 to 1.
6. Evaluate ∫₀^∞ e^(−x) dx.
- ∞
- 1
- 0
- e
Show answer
B: 1. lim (1 − e^(−t)) = 1.
7. As x → ∞, what does x³e^(−x) tend to?
- 0
- ∞
- 1
- 3
Show answer
A: 0. Exponentials dominate powers.
8. How is an improper integral evaluated?
- By ignoring the infinite part
- By numerical estimation only
- As a limit of a proper integral
- It cannot be
Show answer
C: As a limit of a proper integral. Replace ∞ by t and let t → ∞.
9. Does ∫₀¹ 1/x dx converge?
- Yes, to 1
- Yes, to 0
- Yes, to ln 2
- No
Show answer
D: No. −ln t → ∞ as t → 0⁺.
10. What does x ln x tend to as x → 0⁺?
- −∞
- 0
- 1
- ∞
Show answer
B: 0. A standard limit.
Keep going in Full Marks
Full Marks has 53 questions on Improper integrals (43 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Further calculus, plus games, timed exams and past papers.