2021 DHS Promo Q8

1. Using integration by parts, show that \(\displaystyle\displaystyle \displaystyle\int{\frac{1}{{{x}^{2}}}\ln x\mathrm{d}x}=-\frac{1}{x}\ln x-\frac{1}{x}+\beta \), where \(\beta \) is an arbitrary constant.[2]

The graph of the curve \(C\) with equation \(y=\frac{\sqrt{\ln x}-2x}{x}\), \(x\ge 1\) is as shown below. The line \(y=-2\) is its horizontal asymptote.

2. Find the area of the region enclosed by \(C\), the line \(x=4\) and the line \(y=-2x\).[3]
3. The region \(A\) is enclosed by \(C\) and the lines \(x=4\) and \(y=-2\). By using a suitable transformation, show that the exact volume when \(A\) is rotated about the line \(y=-2\) through \(2\pi \) radians is \(\frac{\pi }{4}\left( p\ln 2+q \right)\), where \(p\) and \(q\) are constants to be determined.[4]