It is given that $\mathrm{f}\left( x \right)=\ln \left( 2+2\sin x \right)$. Show that \(

Question

It is given that $\mathrm{f}\left( x \right)=\ln \left( 2+2\sin x \right)$. Show that $\mathrm{f}''\left( x \right)=\frac{k}{1+\sin x}$, where $k$ is a constant to be found.[3] Hence find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{3}}$.[4] Use the series in part (ii) to approximate the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{2}{\mathrm{f}\left( x \right)\,\,\mathrm{d}x}$.[1]

Accepted Answer

(i) $k=-1$ (ii) $f(x)=\ln2+x-\frac12x^2+\frac16x^3+\cdots$ (iii) $2.72$ ($3$ s.f.) (from graphing calculator)

2021 MI P2 Q3

2021 MI P2 Q3

Video Solution:

Solution:

Solution locked