2021 NJC P1 Q6

Using the substitution \(x=\tan \theta \), find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{\frac{1}{\sqrt{1+{{x}^{2}}}}\mathrm{d}x}\).[4]

The graph of \(y=\frac{1}{\sqrt{1+{{x}^{2}}}}\), for \(0\le x\le 1\), is shown in the diagram. Rectangles, each of width \(\frac{1}{n}\), are drawn under the curve.

Show that the total area \(A\) of all \(n\) rectangles is given by

\(A=\frac{1}{\sqrt{{{n}^{2}}+{{1}^{2}}}}+\frac{1}{\sqrt{{{n}^{2}}+{{2}^{2}}}}+\frac{1}{\sqrt{{{n}^{2}}+{{3}^{2}}}}+...+\frac{1}{\sqrt{{{n}^{2}}+{{\left( n-1 \right)}^{2}}}}+\frac{1}{\sqrt{2{{n}^{2}}}}\).

State the limit of \(A\) as \(n\to \infty \).[3]