Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 13 / c

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- Solution c

Solution

A partial fraction decomposition gives

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

For the second term the substitution rule

t = x^{2} + 1

,

d t = 2 x d x

gives a logarithm. Thus

\int \frac{d x}{x ( x ^{2} + 1 )} = ln ∣ x ∣ - \frac{1}{2} ln (x^{2} + 1) + C, x \neq = 0.

The constant in the antiderivative may be chosen independently on the two intervals separated by

0

.

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