Question

If the area bounded by $y={\tan }^{-1}x,y={\cot }^{-1}x$ and $y-$axis is $k$ sq. unit, then the value of $e^k$ is equal to

Answer 1

Required area
$\Delta$=$\underset{0}{\overset{\frac{\pi }{4}}{\int }}\tan ydy+\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}\cot ydy={\ln |\sec y||}_0^{\frac{\pi }{4}}+{\ln |\sin y||}_{\frac{\pi }{4}}^{\frac{\pi }{2}}$


$=\ln \sqrt{2}-\ln \frac{1}{\sqrt{2}}=\ln 2$ sq. units
Therefore, $k=\ln 2$
$\Rightarrow e^k=e^{\ln 2}=2$