2020 TJC P2 Q3

It is given that \(\mathrm{f}(x)=\sin \left[ \ln \left( 1+x \right) \right]\), \(x\in \mathbb{R}\), \(x>-1\).

1. Show that \({{(1+x)}^{2}}\mathrm{f}''(x)+(1+x)\,\mathrm{f}'(x)+\mathrm{f}(x)=0\).[3]
2. By further differentiation of the result in (i), find the first three non-zero terms of the Maclaurin series for \(\mathrm{f}(x)\).[3]
3. Verify the correctness of the series found in part (ii) using the standard series from the List of Formulae (MF26).[2]
4. Use the series in part (ii) to approximate the value of  \(\displaystyle\displaystyle\int_{0}^{2}{\mathrm{f}(x)}\,\mathrm{d}x\).
   Use your calculator to find an accurate value of \(\displaystyle\displaystyle\int_{0}^{2}{\mathrm{f}(x)}\,\mathrm{d}x\). Why is the approximated value not very good?[3]
5. Using the series obtained in part (ii), deduce the Maclaurin series for \(\cos \left[ \ln (1+x) \right]\) in ascending powers of \(x\), up to and including the term in \({{x}^{2}}\).[2]