<p>In this question you may use expansions from the List of Formulae (MF26).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">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.</span></div><p style="text-align: right;">[6]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your expansion from part (i) and integration to find an approximate expression for $\displaystyle\int{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$. Hence find an approximate value for $\displaystyle\int\nolimits_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$ decimal places.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your graphing calculator to find a second approximate value for $\displaystyle\int\nolimits_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$decimal places.</span></div><p style="text-align: right;">[1]</p></li></ol>

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

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.

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 $\displaystyle\displaystyle \displaystyle\int\nolimits_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$ decimal places.

Use your graphing calculator to find a second approximate value for $\displaystyle\displaystyle \displaystyle\int\nolimits_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\mathrm{d}x}$, giving your answer to $4$decimal places.

N2018 P2 Q4

N2018 P2 Q4

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