Question

Solve the equation:-
$\stackrel{lim}{n\to 20}{\int }_{r=0}^{n-1}\frac{n}{n^1+r^2}$

Answer 1

$\underset{n\to 20}{lim}{\int }_r=0^{n-1}\frac{n}{n+r^2}$

$\Rightarrow \underset{n\to 20}{lim}{\int }_r=0^{n-1}\frac{1}{1+(r/\sqrt{n}{)}^2}$

$r/\sqrt{n}=t$

$\frac{1}{\sqrt{n}}\times dr=dt$

$\Rightarrow \sqrt{n}\underset{n\to }{lim}{\int }_0^{n-1}\frac{1}{1+t^2}$

$\Rightarrow \sqrt{n}\underset{n\to 20}{lim}{{}^{}\left[{\tan }^{-1}t\right]}_0^{\frac{n-1}{\sqrt{n}}}$

$\Rightarrow \sqrt{n}\underset{n\to 20}{lim}[{\tan }^{-1}\frac{n-1}{\sqrt{n}}-0]$

$\Rightarrow \sqrt{2}0{\tan }^{-1}\frac{19}{\sqrt{2}0}$