Question

In the interval $\left(0, \frac{\pi}{2}\right)$ area lying between the curves $y = tan\, x$, $y = cot \,x$ and $x$-axis is

• A. $3$ log $2$ sq. units
• B. $2$ log $2$ sq. units
• C. log $2$ sq. units
• D. $\frac{1}{2} $ log $2$ sq. units

Answer 1

Answer: (C)

We have, $y = tan\,x \quad\ldots\left(i\right)$
and $y = cot\,x \quad\ldots\left(ii\right)$
$tan \,x$ and $cot \,x$ intersect at $\pi/ 4$

Required area = area of shaded region
$=\int\limits_{0}^{\pi /4} tan\,x\, dx+\int\limits_{\pi /4}^{\pi/ 2} cot\, x\, dx$
$=\left[-log\, cos\,x\right]_{0}^{\pi/ 4}+\left[log\, sin\,x\right]_{\pi /4}^{\pi /2}$
$=-log \frac{1}{\sqrt{2}}-log \frac{1}{\sqrt{2}}$
$=2\,log \sqrt{2}=log\,2$ sq. units