Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 4 / a

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- Solution a

Solution

Use partial fraction decomposition:

\frac{1}{x ^{2} ( x ^{2} - 4 )} = - \frac{1}{4 x ^{2}} + \frac{1}{4 ( x ^{2} - 4 )} = - \frac{1}{4 x ^{2}} + \frac{1}{16} (\frac{1}{x - 2} - \frac{1}{x + 2}) .

Integrating each term gives the antiderivative

\int \frac{d x}{x ^{2} ( x ^{2} - 4 )} = \frac{1}{4 x} + \frac{1}{16} lo g \frac{x - 2}{x + 2} + C .

This holds on any interval avoiding

- 2, 0, 2

; the constant may differ between these intervals. Differentiation of the logarithm recovers

1/ [4 (x^{2} - 4)]

, providing a direct check.

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