9231 P11 - Nov 2013 - Q4 - 7 marks
6411
It is given that
\(I_n=\int_0^1 \frac{x^n}{\sqrt{1+2x}}\,\mathrm{d}x\).
Show that, for \(n\geqslant 1\),
\((2n+1)I_n=\sqrt3-nI_{n-1}\).
Show that
\(I_3=\frac{2}{35}(\sqrt3+1)\).
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