IB Maths AI HL
Integration
Paper 1 & 2
~4 min read
Introduction to Integration
Integration is differentiation run backwards. Where differentiation takes a function to its gradient, integration recovers the function from its gradient — which is why it’s also called antidifferentiation. The catch: a function and that-same-function-plus-a-constant have identical gradients, so every integral carries a “+ c“.
📘 What you need to know
- Integration is the inverse of differentiation — also called antidifferentiation.
- Antiderivative: the result of integrating; sometimes written F(x).
- Notation: ∫f(x) dx — “∫” means integrate, “dx” means with respect to x.
- Indefinite integral: F(x) = ∫f(x) dx.
- Constant of integration: every indefinite integral ends in “+ c“.
- Why + c? Constants differentiate to 0, so they’re invisible to differentiation — integration must put one back.
The notation
| Symbol | Meaning |
|---|
| ∫ | “integrate” |
| f(x) | the integrand (function being integrated) |
| dx | integrate with respect to x |
| F(x) + c | the antiderivative (indefinite integral) |
You can also think in dydx terms: instead of integrating f(x) to find F(x), integrate dydx to recover y.
The constant of integration
🤔 Why does every integral need “+ c“?
Differentiating a constant gives 0, so x2 + 3, x2 − 7 and plain x2 all share the gradient 2x. Running that backwards, the integral of 2x could be any of them — so we write x2 + c to capture all the possibilities at once. Without extra information, you can’t pin down c.
🧠 “Differentiation forgets the constant — integration puts it back”
The “+ c” is the family of all curves with the same shape, shifted vertically. Each value of c is one member of that family.
💡 Top tips
- Always add “+ c“ for an indefinite integral.
- Read dx to confirm which variable you’re integrating with respect to.
- Think “what differentiates to this?” — that’s the antiderivative.
- One point fixes c — covered in the constant-of-integration topic.
⚠ Common mistakes
- Forgetting “+ c“ — the most common dropped mark in integration.
- Treating integration as unrelated to differentiation — it’s the exact inverse.
- Ignoring the dx — it tells you the variable of integration.
Next up — Integrating Powers of x. Now that integration is set up as the reverse of differentiation, the next topic gives the actual rule for doing it on powers of x: instead of “multiply by the power and drop it by one”, you raise the power by one and divide — the mirror image of the power rule.
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