<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 Maclaurin expansion of $\ln \left( 1+\cos 3x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{4}}$, for $0\le x<\frac{\pi }{3}$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your expansion from part (i) and to find an approximate value for $\int_{0}^{0.5}{x\ln \left( 1+\cos 3x \right)}\,\mathrm{d}x$, giving your answer to $5$ decimal places.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your calculator to find the value of $\int_{0}^{0.5}{x\ln \left( 1+\cos 3x \right)}\,\mathrm{d}x$ up to $5$ decimal places.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">With the aid of a suitable diagram, comparing your answers in (ii) and (iii), comment on the accuracy of your approximations.</span></div><p style="text-align: right;">[2]</p></li></ol>

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

Find the Maclaurin expansion of $\ln \left( 1+\cos 3x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{4}}$, for $0\le x<\frac{\pi }{3}$.

Use your expansion from part (i) and to find an approximate value for $\displaystyle\displaystyle \int_{0}^{0.5}{x\ln \left( 1+\cos 3x \right)}\,\mathrm{d}x$, giving your answer to $5$ decimal places.

Use your calculator to find the value of $\displaystyle\displaystyle \int_{0}^{0.5}{x\ln \left( 1+\cos 3x \right)}\,\mathrm{d}x$ up to $5$ decimal places.

With the aid of a suitable diagram, comparing your answers in (ii) and (iii), comment on the accuracy of your approximations.

2022 SAJC P2 Q3

2022 SAJC P2 Q3

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