2022 NJC P1 Q1
Timothy Gan
2022 NJC P1 Q1
(i)
Given that ${{I}_{n}}=\int_{1}^{\text{e}}{x{{\left( \ln x \right)}^{n}}\,\text{d}x}$ for $n\in \mathbb{Z}$, $n\ge 0$, show that
${{I}_{n}}=\frac{{{\text{e}}^{2}}}{2}-\frac{n}{2}{{I}_{n-1}}$
for all $n\in {{\mathbb{Z}}^{+}}$.
[2]
(ii)
Find the exact volume of the solid generated when the region bounded by the curve $y=\sqrt{x}\ln x$, and the $x$-axis and the line $x=\text{e}$ is rotated completely about the $x$-axis.
[3]
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