Question 0818

Maclaurin Series

2018 Paper 2 Question 4 Variant

Question

In this question you may use expansions from the List of Formulae (MF26).

(i)

Find the Maclaurin expansion of

{\ln(\cos 4 x)}

in ascending powers of

{x,}

up to and including the term in

{x^6.}

State any value(s) of

{x^2}

in the domain

{0 \leq x \leq \frac{1}{8}\pi}

for which the expression is not valid.

[6]

(ii)

Use your expansion in part (i) and integration to find an approximate expression for

\int \frac{\ln(\cos 4 x)}{x^2} \; \mathrm{d}x.

Hence find an approximate value for

\int_0^{0.2} \frac{\ln(\cos 4 x)}{x^2} \; \mathrm{d}x,

giving your answer to 4 decimal places.

[3]

(iii)

Use your graphing calculator to find a second approximate value for

\int_0^{0.2} \frac{\ln(\cos 4 x)}{x^2} \; \mathrm{d}x,

giving your answer to 4 decimal places.

[1]