IB Maths AI SLIntegrationPaper 1 & 2antidifferentiation~6 min read
Introduction to Integration
Differentiation takes a function to its gradient function. Integration runs the process backwards — from a gradient function back to the original. This note covers what integration is, the notation that goes with it, and why every answer carries a mysterious “+ c“.
📘 What you need to know
Integration is the reverse of differentiation — it is also called antidifferentiation.
The result of integrating a function is its antiderivative: a function whose derivative is the one you started with.
The notation ∫f(x) dx means “integrate f(x) with respect to x“; f(x) is the integrand.
The antiderivative is often written F(x), so F(x) = ∫f(x) dx — the indefinite integral of f(x).
Since differentiating a constant gives zero, any constant can be added — so every antiderivative carries a constant of integration, + c.
A single function has endlessly many antiderivatives, one for each value of c.
Integration: reversing differentiation
Differentiation takes a function to its gradient function. Integration runs that process in reverse: starting from the gradient function, it recovers the original. Because it undoes differentiation, integration is also called antidifferentiation, and its result is an antiderivative.
Integration reverses differentiation
if F′(x) = f(x) then F(x) is an antiderivative of f(x)
differentiating an antiderivative takes you straight back to the original function
So if differentiating F(x) gives f(x), then F(x) is an antiderivative of f(x). The quickest way to check any antiderivative is simply to differentiate it and see whether you arrive back at f(x).
Integral notation
Integration has its own notation. The instruction to integrate is written with the long ∫ symbol.
Integral notation
∫ f(x) dx = F(x) + c∫ means integrate · f(x) is the integrand · dx gives the variable · F(x) + c is the indefinite integral
The function being integrated, f(x), is the integrand; the “dx” states that x is the variable of integration. The antiderivative is often named F(x), and the whole expression F(x) + c is the indefinite integral of f(x). The same idea works with dy/dx notation: integrating dy/dx recovers an expression for y.
The constant of integration
Here is the subtlety. Differentiating any constant gives zero — so when you integrate, there is no way to tell whether the original function had a constant term, or what it was. To cover every possibility, an antiderivative always carries a constant of integration, written + c.
Every curve here has the same gradient function — at any x their tangents are parallel (orange). They differ only by the constant c, so a single function has endlessly many antiderivatives.
This means a single function has not one antiderivative but endlessly many — a whole family of curves, all the same shape, each shifted vertically by a different value of c. Without extra information, c cannot be pinned down, so the “+ c” must stay.
🧠Recipe — checking an antiderivative
Identify the function to be integrated, f(x), and the proposed antiderivative, F(x).
Differentiate the proposed antiderivative to find F′(x).
CompareF′(x) with the integrand f(x).
If they match, F(x) is a valid antiderivative — the indefinite integral is F(x) + c.
Remember that F(x) plus any constant is also an antiderivative — this is why every integral carries + c.
Worked examples
WE 1
Checking an antiderivative
Show that F(x) = x5 is an antiderivative of f(x) = 5x4.
to check an antiderivative, differentiate itF(x) = x5F′(x) = 5x4F′(x) = f(x), so x5 is an antiderivative of 5x4integration is checked by differentiation — if differentiating F(x) gives f(x), then F(x) works.
WE 2
Reading integral notation
For the integral ∫(6x2 − 4x) dx: (a) state the integrand; (b) state the variable of integration.
(a) the integrand is the function being integratedintegrand: 6x2 − 4x(b) the “dx” gives the variable of integrationvariable: xthe “dx” is not decoration — it tells you which letter is the variable.
WE 3
Two antiderivatives of one function
Show that both G(x) = x3 + 2 and H(x) = x3 − 7 are antiderivatives of f(x) = 3x2.
differentiate each proposed antiderivativeG′(x) = 3x2 (the +2 differentiates to 0)H′(x) = 3x2 (the −7 differentiates to 0)both derivatives equal f(x) = 3x2, so both are antiderivativesthey differ only by a constant — exactly what the constant of integration allows.
WE 4
Is this an antiderivative?
A student claims that F(x) = x2 is an antiderivative of f(x) = x2. Determine whether this is correct.
check by differentiating the proposed antiderivativeF(x) = x2 ⇒ F′(x) = 2xF′(x) = 2x, which is not equal to f(x) = x2so x2 is not an antiderivative of x2an antiderivative of x2 must have derivative x2 — and x2 differentiates to 2x, not x2.
WE 5
Why the constant is needed
It is given that dy/dx = 4x3, and that the derivative of x4 is 4x3. (a) Write the general antiderivative y. (b) Explain why a constant must be included.
(a) differentiating x4 gives 4x3, so the antiderivative isy = x4 + c(b) differentiating any constant gives 0so x4, x4 + 1, x4 − 5, … all have derivative 4x3(a) y = x4 + c · (b) every value of c gives a valid antiderivativethe “+ c” is not optional — it stands for the whole family of possible antiderivatives.
WE 6
Full question: antiderivatives and the integral
A function is given by F(x) = 2x3 − x + 6. (a) Find F′(x). (b) Hence write down a function f(x) for which F(x) is an antiderivative. (c) Write down two other antiderivatives of that f(x). (d) Write the indefinite integral ∫f(x) dx.
(a) differentiate F(x)F′(x) = 6x2 − 1(b) since F′(x) = 6x2 − 1F(x) is an antiderivative of f(x) = 6x2 − 1(c) any 2x3 − x + (constant) workse.g. 2x3 − x and 2x3 − x + 10(a) 6x2−1 · (b) f(x)=6x2−1 · (c) e.g. 2x3−x, 2x3−x+10 · (d) ∫f(x) dx = 2x3−x+cF(x) is just one member of the family; the indefinite integral with “+ c” represents them all.
💡 Top tips
Integration reverses differentiation — to check any antiderivative, just differentiate it.
Every indefinite integral ends in “+ c” — leaving it out is a guaranteed lost mark.
In ∫f(x) dx, the “dx” names the variable of integration — it is part of the notation, not decoration.
A function has endlessly many antiderivatives; they all differ only by a constant.
F(x) and dy/dx notation describe the same idea — integrate to recover the original function.
âš Common mistakes
Forgetting the constant of integration “+ c” on an indefinite integral.
Thinking a function has only one antiderivative — there is a whole family.
Confusing the integrand with the antiderivative — the integrand is what you start with.
Mixing up the directions — differentiating when the task is to integrate, or the reverse.
Dropping the “dx“ from the notation, or ignoring which variable it specifies.
Next up: Integrating Powers of x — the rule that turns this idea into real calculation: raise the power, then divide by the new power. For now hold the two anchors: integration is differentiation in reverse, and every indefinite integral ends in “+ c“. Forget the c and you forget a mark.
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