Integration Techniques

This lesson covers three key A-Level integration methods: integration by substitution, integration by parts, and integration using partial fractions.

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Integration by Substitution

Substitution reverses the chain rule. Replace a complicated expression with a simpler variable u.

Method

1. Choose a substitution u = g(x).
2. Find du/dx and hence dx in terms of du.
3. Replace all x-terms with u-terms.
4. Integrate with respect to u.
5. Substitute back to x (for indefinite integrals).

Worked Example 1

Find ∫2x(x² + 1)⁴ dx.

Let u = x² + 1, so du = 2x dx.

∫2x(x² + 1)⁴ dx = ∫u⁴ du = u⁵/5 + c

= (x² + 1)⁵/5 + c

Worked Example 2

Find ∫x√(3x − 1) dx.

Let u = 3x − 1, so x = (u + 1)/3 and dx = du/3.

∫[(u + 1)/3] √u × du/3 = (1/9) ∫(u + 1)u^(1/2) du = (1/9) ∫(u^(3/2) + u^(1/2)) du

= (1/9)[2u^(5/2)/5 + 2u^(3/2)/3] + c

= 2(3x − 1)^(5/2)/45 + 2(3x − 1)^(3/2)/27 + c

Definite Integrals with Substitution