2022 SAJC P2 Q3
In this question you may use expansions from the List of Formulae (MF26).
(i)
Find the Maclaurin expansion of $\ln \left( 1+\cos 3x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{4}}$, for $0\le x<\frac{\pi }{3}$.
[4]
(ii)
Use your expansion from part (i) and to find an approximate value for $\int_{0}^{0.5}{x\ln \left( 1+\cos 3x \right)}\,\text{d}x$, giving your answer to $5$ decimal places.
[2]
(iii)
Use your calculator to find the value of $\int_{0}^{0.5}{x\ln \left( 1+\cos 3x \right)}\,\text{d}x$ up to $5$ decimal places.
[1]
(iv)
With the aid of a suitable diagram, comparing your answers in (ii) and (iii), comment on the accuracy of your approximations.
[2]
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