Question

The area bounded by the curve $y={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}x,y$-axis and the line $2y=\pi (x+1)$ is equal to

• A. $\pi$ sq. units
• B. $\frac{\pi }{2}$ sq. units
• C. $\frac{\pi }{3}$ sq. units
• D. $\frac{\pi }{4}$ sq. units

Answer 1

Answer: (B)

$f(x)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}(x)$
$x\in [-1,0)f(x)=\frac{\pi }{2}-\pi +\frac{\pi }{2}=0$
$x\in (0,1]f(x)=\frac{\pi }{2}+\frac{\pi }{2}=\pi$
Required area (shaded region)
$=2\times \frac{1}{2}\times \frac{\pi }{2}\times 1=\frac{\pi }{2}\text{ sq. units. }$