<p>It is given that&nbsp;\(\mathrm{f}(x) = \ln(a + x)\), \(x \in \mathbb{R}\), \(x &gt; -a\), where&nbsp;\(a\) is a constant.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Using the standard series from the List of Formulae (MF27), find the series expansion for&nbsp;\(\mathrm{f}(x)\), up to and including the term in \(x^3\).</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;\(a=1\).</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, show that the series expansion of \(\sin[\mathrm{f}(x)]\), up to and including the term in \(x^3\) is given by \(x - \frac{1}{2}x^2 + \frac{1}{6}x^3\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the Maclaurin series for&nbsp;\(\cos[\mathrm{f}(x)]\) up to and including the term in \(x^2\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find \(\int_{1}^{3} (x - \frac{1}{2}x^2 + \frac{1}{6}x^3) \, \mathrm{d}x\). Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of \(\int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x\).</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that \(\mathrm{f}(x) = \ln(a + x)\), \(x \in \mathbb{R}\), \(x > -a\), where \(a\) is a constant.

Using the standard series from the List of Formulae (MF27), find the series expansion for \(\mathrm{f}(x)\), up to and including the term in \(x^3\).

Hence, or otherwise, show that the series expansion of \(\sin[\mathrm{f}(x)]\), up to and including the term in \(x^3\) is given by \(x - \frac{1}{2}x^2 + \frac{1}{6}x^3\).

Deduce the Maclaurin series for \(\cos[\mathrm{f}(x)]\) up to and including the term in \(x^2\).

Find \(\displaystyle \int_{1}^{3} (x - \frac{1}{2}x^2 + \frac{1}{6}x^3) \, \mathrm{d}x\). Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of \(\displaystyle \int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x\).

2025 VJC P2 Q1

2025 VJC P2 Q1