2023 NJC P2 Q3

Using integration by parts on \(\displaystyle\displaystyle \displaystyle\int{\frac{{{\left( \ln x \right)}^{2}}}{{{x}^{2}}}}\mathrm{d}x\), show that

\(\displaystyle\displaystyle \displaystyle\int{\frac{2\ln x}{{{x}^{2}}}\mathrm{d}x-}\displaystyle\displaystyle \displaystyle\int{\frac{{{\left( \ln x \right)}^{2}}}{{{x}^{2}}}\mathrm{d}x}=\frac{{{\left( \ln x \right)}^{2}}}{x}+c\),

where \(c\) is an arbitrary constant.[3]

Region \(~R\) is bounded by the curves \(y=\frac{\sqrt{2\ln x}}{x}\) and \(y=\frac{\ln x}{x}\). Find the exact volume generated by rotating \(~R\) about the \(x\)-axis through \(2\pi \) radians.[5]