<p>It is given that $\mathrm{f}\left( x \right)=\ln \left( 2+2\sin x \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\mathrm{f}''\left( x \right)=\frac{k}{1+\sin x}$, where $k$ is a constant to be found.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{3}}$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Use the series in part (ii) to approximate the value of $\int_{0}^{2}{\mathrm{f}\left( x \right)\,\,\mathrm{d}x}$.</span></div><p style="text-align: right;">[1]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=\ln \left( 2+2\sin x \right)$.

Show that $\mathrm{f}''\left( x \right)=\frac{k}{1+\sin x}$, where $k$ is a constant to be found.

Hence find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{3}}$.

Use the series in part (ii) to approximate the value of $\displaystyle\displaystyle \int_{0}^{2}{\mathrm{f}\left( x \right)\,\,\mathrm{d}x}$.

2021 MI P2 Q3

2021 MI P2 Q3

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