Partial Fractions in Depth

This lesson covers partial fractions in depth as required by the AQA A-Level Mathematics specification (7357). Partial fractions allow you to decompose a single algebraic fraction into a sum of simpler fractions. This technique is essential for integration, for summing series, and for working with binomial expansions of rational functions.

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Why Partial Fractions?

Many algebraic fractions are difficult to integrate or manipulate in their combined form. By splitting them into simpler fractions, we can:

1. Integrate expressions that would otherwise be intractable.
2. Sum series using the method of differences (telescoping).
3. Expand rational functions using the binomial theorem.

For example, it is not obvious how to integrate 1/((x − 1)(x + 2)), but if we write it as:

1/((x − 1)(x + 2)) = (1/3)/(x − 1) − (1/3)/(x + 2)

then each term can be integrated using the standard result ∫ 1/(x − a) dx = ln|x − a| + C.

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Case 1: Distinct Linear Factors

If the denominator has distinct linear factors, the partial fraction decomposition has the form: