<p>Let $\mathrm{f}\left( x \right)=\frac{x}{\sqrt{x+1}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\int_{0}^{1}{\mathrm{f}\left( x \right)\,\mathrm{d}x=\frac{2}{3}\left( 2-\sqrt{2} \right)}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">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.</div><div class="list-item-paragraph">By considering the total area of these $n$rectangles, deduce that</div><div class="list-item-paragraph" style="text-align: center">$\frac{1}{n\sqrt{n}}\left\{ \frac{1}{\sqrt{n+1}}+\frac{2}{\sqrt{n+2}}+\frac{3}{\sqrt{n+3}}+...+\frac{n}{\sqrt{2n}} \right\}>\frac{2}{3}\left( 2-\sqrt{2} \right)$.<span class="inline-marks">[3]</span></div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1764583702521-gia1jy0h0dv.png" alt="List item image" /></div></li></ol>

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

Show that $\displaystyle\displaystyle \int_{0}^{1}{\mathrm{f}\left( x \right)\,\mathrm{d}x=\frac{2}{3}\left( 2-\sqrt{2} \right)}$.

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

$\frac{1}{n\sqrt{n}}\left\{ \frac{1}{\sqrt{n+1}}+\frac{2}{\sqrt{n+2}}+\frac{3}{\sqrt{n+3}}+...+\frac{n}{\sqrt{2n}} \right\}>\frac{2}{3}\left( 2-\sqrt{2} \right)$.

2023 ACJC P1 Q3

2023 ACJC P1 Q3

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