9231 P23 - Jun 2020 - Q2 - 6 marks
6040
Let \(I_{n}=\int_{0}^{1}(1+3 x)^{n} \mathrm{e}^{-3 x} \mathrm{~d} x\), where \(n\) is an integer.
(a) Show that \(3 I_{n}=1-4^{n} \mathrm{e}^{-3}+3 n I_{n-1}\).
(b) Find the exact value of \(I_{2}\).
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