Using standard series from the List of Formula (MF27), expand $\frac{\cos ax}{x^2 + a^2}$ as far a

Question

Using standard series from the List of Formula (MF27), expand $\frac{\cos ax}{x^2 + a^2}$ as far as the term in $x^4$, where $a$ is a positive constant. Give your answer in the form $c_1 + c_2 x^2 + c_3 x^4$ where $c_1$, $c_2$ and $c_3$ are in terms of $a$.[6] Find the range of values of $x$, in terms of $a$, for which the expansion in part (a) is valid.[1] Using your answer in part (a) and a suitable value of $a$, show that $\int_{0}^{1} \frac{\cos 2x}{x^2 + 4} \,\mathrm{d}x \approx \frac{119}{960}$.[3]

Accepted Answer

(a) $c_1=\dfrac1{a^2}$, $c_2=-\left(\dfrac12+\dfrac1{a^4}\right)$, $c_3=\dfrac{a^2}{24}+\dfrac1{2a^2}+\dfrac1{a^6}$ (b) $-a<x<a$ (c) $\dfrac{119}{960}$

2024 VJC Promo Q9

2024 VJC Promo Q9

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