The area (in square units) bounded by $y= an ^{-1} x, y=\cot ^{-1} x$ and the $Y$-axis, is

Question

The area (in square units) bounded by $y=	an ^{-1} x, y=\cot ^{-1} x$ and the $Y$-axis, is

Aicharya · Accepted Answer

$ \begin{aligned} & 	ext { The required area }=2\left[\frac{\pi}{4}-\int_0^1 	an ^{-1} x d xight] \ & \quad=\frac{\pi}{2}-2 \int_0^1 	an ^{-1} x d x \ & \left.\quad=\frac{\pi}{2}-2\left[x 	an ^{-1} xight)_0^1-\int_0^1 \frac{x}{1+x^2} d xight] \ & =\frac{\pi}{2}-2\left[\frac{\pi}{4}-\frac{1}{2}\left[\log _e\left(1+x^2ight)ight]_0^1ight] \ & =\frac{\pi}{2}-\frac{\pi}{2}+\left[\log _e\left(1+x^2ight)ight]_0^1 \ & =\log _e 2 \end{aligned} $ Hence, option (b) is correct

The area (in square units) bounded by $y=\tan ^{-1} x, y=\cot ^{-1} x$ and the $Y$-axis, is

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