Integration reverses differentiation. The area model shows the antiderivative building up under the curve.
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Integration is the reverse of differentiation: The antiderivative of f(x) is a function F(x) with F′(x) = f(x).
We write ∫ f(x) dx = F(x) + C.
The +C is the constant of integration — any constant disappears when you differentiate, so it must be restored when you integrate.
Example 1: Basic polynomial integration
Find ∫ (3x² + 4x − 1) dx.
Step by step
- Integrate term by term.
- Combine and add + C.
Final answer
∫ (3x² + 4x − 1) dx = x³ + 2x² − x + C.
Do NOT forget + C: Every indefinite integral needs + C.
Differentiating any constant gives 0, so the original function could have had any constant.
Missing + C loses a mark.
What AI SL does (and does not) integrate: In AI SL you only integrate polynomials and powers of x with integer n ≠ −1 (including negative powers like x⁻²).
You will NOT integrate 1/x (ln), eˣ, sin x or cos x, and you do NOT use integration by substitution — those belong to a different course.
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Rewrite first, then integrate
Turn fractions and brackets into powers of x: The power rule needs each term as a single power of x.
Rewrite fractions as negative powers, and expand brackets, BEFORE integrating.
Example: negative powers
Find ∫ (2/x³ + 5) dx.
Step by step
- Rewrite the fraction as a power with an integer exponent.
- Integrate 2x⁻³ with the power rule.
- Integrate the constant 5.
Final answer
∫ (2/x³ + 5) dx = −1/x² + 5x + C.
Example: expand brackets first
Find ∫ x(x − 4) dx.
Step by step
- Expand into a sum of powers.
- Integrate term by term, add + C.
Final answer
∫ x(x − 4) dx = x³/3 − 2x² + C.
You cannot integrate a product directly: There is no product rule for integration in AI SL.
Always expand brackets (or split a fraction) into separate powers of x first.
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Integrating term by term and checking
Each term separately: Integrate each term on its own.
The coefficient stays; apply the power rule to the power of x.
Add a single + C at the end.
Example: a longer polynomial
Find ∫ (6x² − 4x + 7) dx.
Step by step
- Integrate each term.
- Combine and add + C.
Final answer
∫ (6x² − 4x + 7) dx = 2x³ − 2x² + 7x + C.
Verify by differentiating: You can always check an integral by differentiating your answer — it should give back the original integrand.
Differentiating 2x³ − 2x² + 7x + C gives 6x² − 4x + 7.
Correct.
Finding the constant from a known point
A point pins down + C: Integration gives F(x) + C, a whole family of curves.
If you know one point on the curve, substitute it to find C.
(This is covered fully in the next topic, 5.5.3.)
Example: from gradient to function
A curve has gradient f′(x) = 3x² − 2 and passes through (1, 4).
Find f(x).
Step by step
- Integrate the gradient.
- Use f(1) = 4 to find C.
Final answer
f(x) = x³ − 2x + 5.
Example: total amount from a rate
Water flows into a tank at a rate dV/dt = 3t² litres per minute.
At t = 0 the tank holds 10 litres.
Find V(t).
Step by step
- Integrate the rate.
- Use V(0) = 10.
Final answer
V(t) = t³ + 10 litres.