IB Maths AA SL Topic 5 — Calculus Paper 1 & 2 ~6 min read

Introduction to Integration

Integration is just differentiation in reverse. If differentiating gives you the gradient function, integrating brings you back to the original function. That’s the whole idea.

📘 What you need to know

Integration = the inverse of differentiation. Also called antidifferentiation.
The result is called the antiderivative, often written F(x).
Notation: ∫ f(x) dx = F(x) + c.
Always add “+ c“ — the constant of integration.
Without extra info, you can’t pin down the value of c. Many curves work — they’re all parallel.

The big idea: integration undoes differentiation

The two-way street

x³ differentiate → 3x²

x³ + c integrate ← 3x²

If you know how to differentiate, you can integrate — just run the steps backwards. If differentiating x³ gives 3x², then integrating 3x² gives x³ (plus a constant).

The notation

What each part means

∫ f(x) dx “integrate” function to integrate w.r.t. x

So ∫ f(x) dx reads as “integrate f(x) with respect to x“. You can also use dy/dx notation: integrating dydx gives back y.

The constant of integration “+ c”

Every time you integrate, you add a “+ c” at the end. Why?

🤔 Why “+ c”?

Differentiating any constant gives 0. So if f(x) = x² + 7, its derivative is 2x. But so is the derivative of x² + 100, or x² − 4. When you integrate 2x, you can’t tell which constant was originally there — so you write “+ c” to cover all possibilities.

Indefinite integral

∫ f(x) dx = F(x) + c

🧠

“Indefinite = no limits = needs c”

An indefinite integral has no limits and gives a family of curves → always add + c. A definite integral (later) has limits and gives a number → no c needed.

A family of curves

Different values of c give you different curves — all the same shape, just shifted up or down.

y = x² + c for different values of c

📍

One point pins down the curve

If a question gives you one point on the curve, you can find c by substituting in. We’ll cover this fully in “Finding the Constant of Integration”.

Worked examples

WE 1

Verify an antiderivative by differentiating

Show that F(x) = x³ + 5 is an antiderivative of f(x) = 3x².

Differentiate F(x) and check it equals f(x). F(x) = x³ + 5 F′(x) = 3x² + 0 = 3x² F′(x) = f(x) ✓ F(x) = x³ + 5 IS an antiderivative of 3x² notice — F(x) = x³ + 100 would also work! the +5 was just one possible “c”.

WE 2

Find an antiderivative by reversing

Find ∫ 4x³ dx by thinking about what differentiates to give 4x³.

Ask: “What did I differentiate to get 4x³?” d/dx(x⁴) = 4x³ ✓ So: ∫ 4x³ dx = x⁴ + c ∫ 4x³ dx = x⁴ + c don’t forget the +c! it’s not optional.

WE 3

List several antiderivatives of the same function

Write down three different antiderivatives of f(x) = 2x.

Any function of the form x² + c is an antiderivative of 2x. Pick three values of c.c = 0: F(x) = x² c = 1: F(x) = x² + 1 c = −5: F(x) = x² − 5 All three differentiate to give 2x ✓ x², x² + 1, x² − 5 (any choice of c works) infinitely many antiderivatives — they’re all parallel curves!

WE 4

Identify the parts of an integral

For the integral ∫ (3x² + 5) dx, identify the integrand and the variable of integration.

Integrand (the function inside): 3x² + 5 Variable of integration (from “dx”): x integrand: 3x² + 5 · variable: x “dx” tells you which letter you’re integrating against — important when there are several variables!

💡 Top tips

Always add “+ c” to indefinite integrals. Forgetting it loses easy marks.
Check your answer by differentiating it — you should get back the original function.
Integration is undoing differentiation — work backwards through the rules you already know.
“dx” tells you the variable. ∫ f(x) dx = with respect to x. ∫ f(t) dt = with respect to t.
Don’t confuse F(x) and f(x) — capital F is the antiderivative, lowercase f is the original.

⚠ Common mistakes

Forgetting “+ c” — the most common error in indefinite integrals.
Trying to differentiate when the question says integrate. They’re opposites!
Ignoring “dx” — without it, the integral isn’t defined.
Treating integration as a separate skill — it’s literally differentiation in reverse. If you can do one, you can do the other.
Saying “+ c” but forgetting to use it later when more info is given.

Now you know what integration is. The next note shows the actual rule for integrating powers of x — and how it’s just the power rule running backwards.

Need help with Introduction to Integration?

Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.

Book Free Session →