Home page for accesible maths 1 1 Further Integration 1.1 Rational functions 1.3 Dividing polynomials

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1.2 Systematic integration of rational functions

Theorem.

Suppose that $f(x)$ and $g(x)$ are polynomials such that we can find the roots of the equation $g(x)=0$ . Then we can find the indefinite integral

\int{\frac{f(x)}{g(x)}}dx

by division, partial fractions, and substitutions.

This gives a systematic approach to integrating $\int h(x)dx$ , which will be outlined here. The technique involves reduction to special cases that can be dealt with by integration by substitution. The crucial cases to consider are when $g$ is a linear factor or a quadratic. The outcome involves inverse tangents, logarithms and rational functions.