Question

In the interval $\left(0,\frac{\pi }{2}\right)$ area lying between the curves $y=tanx$, $y=cotx$ 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, $y=tanx\dots \left(i\right)$
and $y=cotx\dots \left(ii\right)$
$tanx$ and $cotx$ intersect at $\pi /4$

Required area = area of shaded region
$=\underset{0}{\overset{\pi /4}{\int }}tanxdx+\underset{\pi /4}{\overset{\pi /2}{\int }}cotxdx$
$={\left[-logcosx\right]}_0^{\pi /4}+{\left[logsinx\right]}_{\pi /4}^{\pi /2}$
$=-log\frac{1}{\sqrt{2}}-log\frac{1}{\sqrt{2}}$
$=2log\sqrt{2}=log2$ sq. units