Integration Using Partial Fractions

This lesson covers integrating rational functions using partial fraction decomposition — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to decompose fractions into simpler parts and integrate each part, often producing logarithmic results.

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Why Partial Fractions?

A rational function is a fraction where both the numerator and denominator are polynomials. Many rational functions cannot be integrated directly, but when decomposed into partial fractions, each part can be integrated easily.

For example:

∫ 1/[(x - 1)(x + 2)] dx is difficult as it stands.

But 1/[(x - 1)(x + 2)] = (1/3)/(x - 1) - (1/3)/(x + 2)

Now each fraction can be integrated using the standard result ∫ 1/(x - a) dx = ln|x - a| + c.

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Review of Partial Fraction Decomposition

Type 1: Distinct Linear Factors

If the denominator has distinct (different) linear factors:

f(x) / [(x - a)(x - b)] = A/(x - a) + B/(x - b)

Example: Express 5x + 1 / [(x + 1)(x - 2)] in partial fractions.

5x + 1 / [(x + 1)(x - 2)] = A/(x + 1) + B/(x - 2)