N2018 P2 Q4

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

1. Find the Maclurin expansion of \(\ln (\cos 2x)\) in ascending powers of \(x\), up to and including the term \({{x}^{6}}\). State any value(s) of \(x\) in the domain \(0\le x\le \frac{1}{4}\pi \) for which the expansion is not valid.[6]
2. Use your expansion from part (i) and integration to find an approximate expression for \(\displaystyle\displaystyle \displaystyle\int{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}\). Hence find an approximate value for \(\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}\), giving your answer to \(4\) decimal places.[3]
3. Use your graphing calculator to find a second approximate value for \(\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}\), giving your answer to \(4\)decimal places.[1]