Topic 1 — Fundamental Theorem of Calculus
What does it actually say?
The Fundamental Theorem of Calculus makes a single profound claim: integration and differentiation are inverse operations. If you differentiate a function and then integrate the result, you get back to where you started (plus a constant). If you integrate and then differentiate, you also get back.
🧠 Analogy — the zoom-in/zoom-out idea: Think of differentiation as zooming in on a curve — it tells you the slope at every point. Integration is zooming back out — it rebuilds the curve from the slope information. They undo each other perfectly, like squaring and taking a square root.
y = f(x) + c
↑ INTEGRATION DIFFERENTIATION ↓
dy/dx = f ′(x)
The two operations are perfect inverses — differentiation destroys the constant, integration cannot recover it without extra information.
This particular form of the theorem — relating to building a function back from its derivative — underpins indefinite integration. (There is also a definite-integration version, which you'll meet when computing areas.)
The Constant of Integration — why "+c" is not optional
When you differentiate any constant (like 5, or −100, or π), the answer is always 0. Constant terms completely vanish under differentiation:
This creates a problem when running in reverse. If you know that dy/dx = 5, you can tell that y = 5x... but also that y = 5x + 2, or y = 5x − 7 — all three have the same derivative. There are infinitely many functions that differentiate to give 5.
To be honest about this unknown, we write +c (called the constant of integration). It represents every possible constant that could be there. We are saying: "I know the shape of this curve, but I don't know how high up it sits until I'm given a point."
Never forget +c in indefinite integration. Every step of your working must carry it forward once you've integrated. Examiners will deduct marks for a missing constant of integration even if everything else is perfect.
Notation — reading the integral sign
The notation can look intimidating at first. Let's decode it piece by piece:
∫ (3x² + 5x + 4) dx
∫ = integral sign (stretched-S, meaning "sum")
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(3x² + 5x + 4) = the integrand (what you're integrating)
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dx = "with respect to x"
"Integrate the whole of (3x² + 5x + 4) with respect to x"
- The ∫ sign always pairs with a dx (or dy, dt, etc.). Think of them as opening and closing brackets around what you're integrating.
- The dx tells you the variable — any other letter in the expression is treated as a constant.
Interpreting dx:
t
Worked Example