- (a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos ^ { 2 } \left( \frac { x } { 3 } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\)
Give each term in simplest form.
(b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
(c) Use the integration function on your calculator to evaluate
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
Give your answer to 5 decimal places.
(d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).