Integration by Substitution

This lesson covers integration by substitution — a powerful technique for integrating composite functions — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to be able to choose an appropriate substitution, change variables (and limits for definite integrals), and recognise patterns that suggest substitution.

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Why Substitution?

Many integrals cannot be evaluated directly using the basic rules. For example:

∫ 2x(x² + 1)⁵ dx

You cannot expand (x² + 1)⁵ easily. Instead, we use substitution to transform the integral into something simpler.

The method is the reverse of the chain rule for differentiation.

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The Method

Step-by-Step Process

1. Choose a substitution u = g(x) (usually the "inner function" of a composite).
2. Find du/dx and rearrange to express dx in terms of du.
3. Replace all x-terms in the integral with expressions involving u.
4. Integrate with respect to u.
5. Substitute back to express the answer in terms of x (for indefinite integrals).

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Worked Examples — Indefinite Integrals

Example 1: ∫ 2x(x² + 1)⁵ dx

Step 1: Let u = x² + 1

Step 2: du/dx = 2x, so du = 2x dx