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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.
Many algebraic fractions are difficult to integrate or manipulate in their combined form. By splitting them into simpler fractions, we can:
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.
If the denominator has distinct linear factors, the partial fraction decomposition has the form:
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