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

Question

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 $x^3$. Use your result from part (ii) to approximate the value of $\displaystyle \displaystyle\int_{0}^{1} \frac{e^{\sin x}}{\sqrt{1+2x}} \, \mathrm{d}x$. 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 $x^3$.

Accepted Answer

(i) Identity shown. (ii) $y=1+x^2-\frac32x^3+\cdots$. (iii) $\frac{23}{24}\approx0.958$; poor approximation because the cubic differs substantially near $x=1$. (iv) $1+x^2+\frac32x^3+\cdots$. Both expansions are valid for $|x|<\frac12$.

2019 HCI P2 Q2

2019 HCI P2 Q2

Solution:

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