Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 3

Table of contents
- a 3
  - Solution a
- b 3
  - Solution b

a

Solution

Since the derivative of

ln cos x

is

- tan x

on this interval,

\int_{0}^{π /4} tan x d x = [- ln cos x]_{0}^{π /4} = - ln \frac{1}{2} = \frac{1}{2} ln 2 .

b

Solution

Use the real antiderivative

ln ∣ x ∣

, valid on either half-line away from zero:

\int_{- 2}^{- 1} \frac{d x}{x} = [ln ∣ x ∣]_{- 2}^{- 1} = ln 1 - ln 2 = - ln 2 .

The negative answer is consistent with a negative integrand and increasing integration limits.

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