 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/5/a/solution

The double-angle identity for the cosine gives

cos^{2} x - 1 = (cos 2 x - 1) /2

. Termwise integration therefore gives

\int (cos^{2} x - 1) d x = \frac{1}{4} sin 2 x - \frac{x}{2} + C .

Here

C

is an arbitrary constant in the antiderivative.

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