<p>It is given that \(y = \frac{e^{\sin x}}{\sqrt{1+2x}}\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that \(\frac{1}{y} \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{1}{1+2x} = \cos x\).</span></div></li><li><div class="list-item-paragraph">By further differentiation of the result in part <strong>(i)</strong>, find the Maclaurin series for \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\).</div></li><li><div class="list-item-paragraph">Use your result from part <strong>(ii)</strong> to approximate the value of \(\displaystyle \displaystyle\int\nolimits_{0}^{1} \frac{e^{\sin x}}{\sqrt{1+2x}} \, \mathrm{d}x\). Explain why this approximation obtained is not good.</div></li><li><div class="list-item-paragraph">Deduce the Maclaurin series for \(\frac{1}{e^{\sin x} \sqrt{1-2x}}\) in ascending powers of \(x\), up to and including the term in \(x^3\).</div></li></ol>

It is given that \(y = \frac{e^{\sin x}}{\sqrt{1+2x}}\).

Show that \(\frac{1}{y} \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{1}{1+2x} = \cos x\).

By further differentiation of the result in part (i), find the Maclaurin series for \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\).

By further differentiation of the result in part **(i)**, find the Maclaurin series for \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\).

Use your result from part (ii) to approximate the value of \(\displaystyle \displaystyle\int\nolimits_{0}^{1} \frac{e^{\sin x}}{\sqrt{1+2x}} \, \mathrm{d}x\). Explain why this approximation obtained is not good.

Use your result from part **(ii)** to approximate the value of \(\displaystyle \displaystyle \displaystyle\int\nolimits_{0}^{1} \frac{e^{\sin x}}{\sqrt{1+2x}} \, \mathrm{d}x\). Explain why this approximation obtained is not good.

Deduce the Maclaurin series for \(\frac{1}{e^{\sin x} \sqrt{1-2x}}\) in ascending powers of \(x\), up to and including the term in \(x^3\).

2019 HCI P2 Q2

2019 HCI P2 Q2