Question

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

• A. $\int\limits_0^{\pi / 4} \tan x d x+\int\limits_{\pi / 4}^{\pi / 2} \cot x d x$
• B. $\int\limits_0^{\pi / 4} \tan ^{-1} x d x+\int\limits_{\pi / 4}^{\pi / 2} \cot ^{-1} x d x$
• C. $\frac{-2}{\pi} \int\limits_0^{\pi / 2} \ln (\sin x) d x$
• D. $\int\limits_0^{\pi / 4} \cot ^{-1} x d x+\int\limits_{\pi / 4}^{\pi / 2} \tan ^{-1} x d x$

Answer 1

Answer: (A), (C)

Area $=\int\limits_0^{\pi / 4}(\tan y) d y+\int\limits_{\pi / 4}^{\pi / 2}(\cot y) d y=\ln 2=\frac{-2}{\pi} \int\limits_0^{\pi / 2} \ln (\sin x) d x$