Q. The area bounded by $y = \tan^{- 1}x,\ \ x = 1$ and $x$-axis is
A) $\left( \frac{\pi}{4} + \log\sqrt{2} \right)\ sq\ unit$
B) $\left( \frac{\pi}{4} - \log\sqrt{2} \right)\ sq\ unit$
C) $\left( \frac{\pi}{4} - \log\sqrt{2} + 1 \right)\ sq\ unit$
D) None of these
Explanation
(b)
Required area $=$ Area of rectangle $OABC -$Area of curve $OABO$
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$\frac{\pi}{4} - \int_{0}^{\pi/4}{\tan{y\ dy}}$
$= \frac{\pi}{4} + \lbrack{\log{\cos{y\rbrack}}}_{0}^{\pi/4}$
$= \frac{\pi}{4} + \log{\cos{\frac{\pi}{4} - \log{\cos{(0)}}}}$
$= \frac{\pi}{4} + \log{1 - \log{\sqrt{2} - \log 1}}$
$= \left( \frac{\pi}{4} - \log\sqrt{2} \right)\ sq\ unit$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.