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  4. The area bounded by the curve y= sin -1 x+ cos -1 x+ tan -1((1/x))+ tan -1 x, y-axis and the line 2 y =π( x +1) is equal to

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Q. 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 $2 y =\pi( x +1)$ is equal to

Application of Integrals

Solution:

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