Integration
What integration is
Integration is the reverse of differentiation. It is used to find an original function from its gradient function, and to find the area under a curve.
The reverse power rule
For ∫ axⁿ dx (n ≠ −1):
∫ axⁿ dx = (a ÷ (n+1)) x^(n+1) + c"Add one to the power, divide by the new power." Always include the constant of integration + c for indefinite integrals (because the derivative of a constant is 0).
- Example:
∫ 6x² dx = 2x³ + c. ∫ 1/x dx = ln|x| + c(the special case n = −1).
Standard integrals
| ∫ | Result | ||
|---|---|---|---|
| eˣ | eˣ + c | ||
| 1/x | ln | x | + c |
| cos x | sin x + c | ||
| sin x | −cos x + c |
Finding c
If you know a point the curve passes through, substitute it after integrating to find c.
Definite integration — area under a curve
A definite integral has limits and gives a numerical value (the area between the curve and the x-axis):
∫[a to b] f(x) dx = F(b) − F(a)where F is the integral of f. No + c is needed (it cancels).
- Area below the x-axis comes out negative — split the integral at the roots and take the magnitude if you want total area.
- Area between two curves = ∫(top − bottom) dx over the region.
Worked example
Evaluate ∫[1 to 3] (2x) dx.
- Integrate: x². Evaluate: [x²] from 1 to 3 = 3² − 1² = 9 − 1 = 8. ✓
Common mistakes
- Forgetting + c on indefinite integrals.
- Using the power rule when n = −1 (that integrates to ln|x|).
- Ignoring negative areas below the x-axis when finding total area.
Exam tips
- Rewrite in index form, then "add one to the power, divide by the new power".
- For definite integrals, evaluate F(top) − F(bottom).
- Sketch the curve to spot areas below the axis and split the integral at roots.
Key facts to remember
- Integration reverses differentiation: ∫axⁿ dx = (a/(n+1))x^(n+1) + c (n ≠ −1); ∫1/x dx = ln|x| + c.
- Always add + c for indefinite integrals; find c from a known point.
- Definite integral ∫[a→b] = F(b) − F(a) = area under the curve (negative below the axis).