Question

Area bounded by $y={\tan }^{-1}x,y={\cot }^{-1}x$ and $y-$axis is equal to

• A. $\ln \sqrt{2}$ sq. units
• B. $\ln 4$ sq. units
• C. $\ln 8$ sq. units
• D. $\ln 2$ sq. units

Answer 1

The correct option is D $\ln 2$ sq. units

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