14.10 Integration Concept

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Understand integration as the reverse process of differentiation
  • Know that solutions for indefinite integrals must always include an arbitrary constant

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What is Integration?

You already know how to differentiate — that means starting with an equation for y and finding dy/dx (the gradient function).

Integration is the opposite. It means starting with dy/dx and working backwards to find y.

Because of this, integration is often called the reverse process of differentiation, or the anti-derivative.

🔁 Think of it like this: Differentiation is like tying your shoelaces. Integration is like untying them — it undoes what differentiation did.

Why Do We Need Integration?

Suppose someone tells you that the gradient of a curve at any point is:

dydx=2x\frac{dy}{dx} = 2x

You might ask: "What is the equation of the curve itself?"

That is exactly what integration answers. You work backwards from the gradient function to find the original equation.

The Basic Rule: Reversing Differentiation

Recall the rule for differentiation of a power:

If y=xny = x^n, then dydx=nxn−1\dfrac{dy}{dx} = nx^{n-1}

To reverse this (i.e. to integrate), you do the opposite steps:

Increase the power by 1, then divide by the new power.

This gives the Power Rule for Integration:

∫xn dx=xn+1n+1+c(where n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + c \qquad \text{(where } n \neq -1\text{)}

The symbol ∫…dx\int \ldots dx means "integrate with respect to x". It is the instruction to reverse-differentiate.

📝 Step-by-step breakdown of the rule:

  1. Take the power of x
  2. Add 1 to the power
  3. Divide the whole term by the new power
  4. Always add + c at the end

What is the Arbitrary Constant c?

This is one of the most important ideas in this topic, so read carefully.

When you differentiate, the constant term disappears. For example:

Original curveAfter differentiating
y=x2+5y = x^2 + 5dydx=2x\dfrac{dy}{dx} = 2x
y=x2+2y = x^2 + 2dydx=2x\dfrac{dy}{dx} = 2x
y=x2y = x^2dydx=2x\dfrac{dy}{dx} = 2x
y=x2−3y = x^2 - 3dydx=2x\dfrac{dy}{dx} = 2x

All four different curves give exactly the same gradient function: 2x2x.

This means that when you reverse the process (integrate), you cannot know what the constant was. It could have been +5+5, +2+2, 00, −3-3, or any number at all.

To show this, we write + c at the end of every answer, where c stands for some unknown constant number.

The letter c is called the arbitrary constant. "Arbitrary" simply means it can be any number — we don't know its value unless we are given extra information.

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