2025 MI P1 Q8

It is given that \(y = \mathrm{e}^{2\sin^{-1} x}\).

1. Show that \(\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y\).[2]

2. By further differentiation of the result in part (i), find the Maclaurin series for \(y\) up to and including the term in \(x^2\).

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3. Using your result from part (ii), find an approximate value for \(\displaystyle \displaystyle\int_{0}^{0.1} x\mathrm{e}^{2\sin^{-1} x} \, \mathrm{d}x\), giving your answer to \(4\) significant figures.[2]