2022TrialShort answer

Q12 (a) (ii)

2 marks

Question

Hence, or otherwise, find
$\int\frac{x^2-2x}{x^2-2x-15}\,dx.$

Try the question before revealing the answer.

Solution

\int \frac{x^2-2x}{x^2-2x-15}\,dx =\int \frac{x^2-2x-15+15}{x^2-2x-15}\,dx

=\int dx+\int \frac{15}{x^2-2x-15}\,dx

=\int dx-\frac{15}{8}\int\frac{1}{x+3}\,dx+\frac{15}{8}\int\frac{1}{x-5}\,dx

=x+\frac{15}{8}\left[\ln(x-5)-\ln(x+3)\right]+C

=x+\frac{15}{8}\ln\left(\frac{x-5}{x+3}\right)+C

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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