Question

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

• A. $\underset{0}{\overset{\pi /4}{\int }}\tan xdx+\underset{\pi /4}{\overset{\pi /2}{\int }}\cot xdx$
• B. $\underset{0}{\overset{\pi /4}{\int }}{\tan }^{-1}xdx+\underset{\pi /4}{\overset{\pi /2}{\int }}{\cot }^{-1}xdx$
• C. $\frac{-2}{\pi }\underset{0}{\overset{\pi /2}{\int }}\ln (\sin x)dx$
• D. $\underset{0}{\overset{\pi /4}{\int }}{\cot }^{-1}xdx+\underset{\pi /4}{\overset{\pi /2}{\int }}{\tan }^{-1}xdx$

Answer 1

Answer: (A), (C)

Area $=\underset{0}{\overset{\pi /4}{\int }}(\tan y)dy+\underset{\pi /4}{\overset{\pi /2}{\int }}(\cot y)dy=\ln 2=\frac{-2}{\pi }\underset{0}{\overset{\pi /2}{\int }}\ln (\sin x)dx$