Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 5

Table of contents
- a 5
  - Solution a
- b 5
  - Solution b

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.

b

Solution

The derivative of the tangent is

sec^{2} x

, and the order of the endpoints must be retained. Thus the definite integral is

[tan x]_{π /3}^{π /6} = \frac{1}{3} - 3 = - \frac{2}{3} .

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