2023 ACJC P1 Q3

Let \(\mathrm{f}\left( x \right)=\frac{x}{\sqrt{x+1}}\).

1. Show that \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{\mathrm{f}\left( x \right)\,\mathrm{d}x=\frac{2}{3}\left( 2-\sqrt{2} \right)}\).[2]
2. The diagram below shows a sketch of the curve with equation \(y=\mathrm{f}\left( x \right)\). Rectangles each of width \(\frac{1}{n}\) are drawn as shown.
   
   By considering the total area of these \(n\) rectangles, deduce that
   \(\frac{1}{n\sqrt{n}}\left\{ \frac{1}{\sqrt{n+1}}+\frac{2}{\sqrt{n+2}}+\frac{3}{\sqrt{n+3}}+...+\frac{n}{\sqrt{2n}} \right\}>\frac{2}{3}\left( 2-\sqrt{2} \right)\).[3]