Given that ${{I}_{n}}=\displaystyle\int_{1}^{\mathrm{e}}{x{{\left( \ln x \right)}^{n}}\,\mathrm{d}x}

Question

Given that ${{I}_{n}}=\displaystyle\int_{1}^{\mathrm{e}}{x{{\left( \ln x \right)}^{n}}\,\mathrm{d}x}$ for $n\in \mathbb{Z}$, $n\ge 0$, show that ${{I}_{n}}=\frac{{{\mathrm{e}}^{2}}}{2}-\frac{n}{2}{{I}_{n-1}}$ for all $n\in {{\mathbb{Z}}^{+}}$. [2] 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 $x=\mathrm{e}$ is rotated completely about the $x$-axis. [3]

Accepted Answer

(ii) $\frac{\pi }{4}\left( {{\mathrm{e}}^{2}}-1 \right)$ units$^{3}$

2022 NJC P1 Q1

2022 NJC P1 Q1

Video Solution:

Solution:

Solution locked