Key Idea: Integration is differentiation in reverse — it recovers a function from its rate of change and measures the area under a curve. On Paper 1 you integrate by hand, so the power rule and the + C habit are everything.
∫ The power rule for integration
- the power — add 1 to it, then divide by the new power
- the constant of integration — always there on an indefinite integral
Integration undoes differentiation: add 1 to the power, divide by the new power, then add + C. Integrate a polynomial term by term with one + C at the end. A constant like 5 integrates to 5x (think 5 = 5x⁰).
🧩 Indefinite vs definite
| Indefinite ∫f(x) dx | Definite ∫ₐᵇ f(x) dx |
|---|---|
| Gives a function + C | Gives a number — no + C |
| Answer: F(x) + C | Answer: F(b) − F(a) (top minus bottom) |
| Use a point to pin down C | For f ≥ 0 it equals the area under the curve |
- the lower and upper limits (the x-values bounding the region)
Tip: Turn roots and fractions into powers before integrating: $\sqrt{x} = x¹/²$ and $1/x² = x⁻²$. Dividing by 3/2 means × 2/3. The rule excludes n = −1 (that case is logs, Topic 5.x).
✏️ IB-style worked examples
IB-style question — integrate, then find f from a point
(a) Find ∫(9x² − 4x + 7) dx. (b) Find ∫(√x + 1/x²) dx. (c) A curve has f′(x) = 6x² − 2 and passes through (1, 5). Find f(x).
Step by step:
(a) Integrate term by term, one + C at the end.
(b) Rewrite as powers first: √x = x¹/², 1/x² = x⁻².
(c) Integrate f′, then use f(1) = 5 to find C.
(a) 3x³ − 2x² + 7x + C (b) (2/3)x³/² − 1/x + C (c) f(x) = 2x³ − 2x + 5
IB-style question — evaluate, then find a limit from a given area
(a) Evaluate ∫₁⁴ 2x dx. (b) The area under y = 2x from x = 0 to x = k is 16 (k > 0). Find k.
Step by step:
(a) Integrate (no + C), then do F(top) − F(bottom).
(b) Set the definite integral equal to the area.
(b) Solve and keep the sensible (positive) root.
(a) 15 (b) k = 4
Important: An indefinite integral always needs + C — leaving it off loses a mark. And when you're given a point, the work isn't done until you've used it to find C — don't leave the answer as … + C.
Tap each card to reveal the answer.
∫x⁵ dx x⁶/6 + C — add 1 to the power, divide, + C.
∫(4x³ + 1) dx x⁴ + x + C — term by term; the 1 integrates to x.
∫(1/x²) dx −1/x + C — that's x⁻², so add 1 → −1 and divide.
Why no + C on ∫₂⁵ f dx? It's a definite integral — it gives a number, F(5) − F(2).
f′(x) = 4x, f(0) = 3. Find f(x). f(x) = 2x² + 3 — integrate, then f(0) = 3 gives C = 3.
Evaluate ∫₀² 3x² dx 8 — integrate to x³, then 2³ − 0³ = 8.
Exam Tips
- Power rule: add 1 to the power, divide by the new power, + C.
- Indefinite → + C (and find C from a point if one is given); definite → a number, no + C.
- Rewrite roots and fractions as powers (√x = x¹/², 1/xⁿ = x⁻ⁿ) before integrating.
- Definite integral = F(top) − F(bottom); reversing the limits flips the sign.
- Area given? Set the definite integral equal to it and solve for the unknown limit (keep the sensible root).