Integration

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Extend reverse differentiation to include the integration of e^(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b) and sec²(ax + b)

  2. Use trigonometric relationships in carrying out integration (e.g., using double-angle formulae to integrate sin²x or cos²(2x))

  3. Understand and use the trapezium rule to estimate the value of a definite integral, including determining whether the rule gives an over-estimate or under-estimate

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What is Integration?

Integration is the reverse process of differentiation. When we differentiate a function, we find its rate of change. When we integrate, we work backwards to find the original function.

Think of it this way:

  • Differentiation asks: "What is the slope/gradient?"
  • Integration asks: "What function has this slope/gradient?"

Because integration is the reverse of differentiation, we call it reverse differentiation or anti-differentiation.

The Constant of Integration

When we integrate, we always add a constant (usually written as + C) at the end. This is called the constant of integration.

Why do we need it?

When you differentiate any constant number, you get zero. For example:

  • d/dx(5) = 0
  • d/dx(100) = 0
  • d/dx(C) = 0

So when we reverse this process (integrate), we don't know what the original constant was. It could have been any number! That's why we write + C to represent "some unknown constant."

Example:

  • If d/dx(x² + 5) = 2x
  • And d/dx(x² + 100) = 2x
  • And d/dx(x² + C) = 2x

Then: ∫2x dx = x² + C

Objective 1: Standard Integration Results

These are the key integration formulas you need to know. Each one comes directly from reversing the differentiation rules.

1.1 Integrating e^(ax + b)

Formula: ∫eax+b dx=1aeax+b+C\int e^{ax+b} \, dx = \frac{1}{a} e^{ax+b} + C

How to remember this: When you differentiate e^(ax+b), you get a × e^(ax+b) (because of the chain rule). So when integrating, you divide by that coefficient 'a'.

Example 1: Integrate e^(3x + 2)

Solution:

  • Here, a = 3 and b = 2
  • ∫e^(3x+2) dx = (1/3)e^(3x+2) + C

Example 2: Integrate e^(5x)

Solution:

  • Here, a = 5 and b = 0
  • ∫e^(5x) dx = (1/5)e^(5x) + C

Example 3: Integrate e^(-2x + 1)

Solution:

  • Here, a = -2 and b = 1
  • ∫e^(-2x+1) dx = (1/-2)e^(-2x+1) + C = -(1/2)e^(-2x+1) + C

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