Question

In the interval $[0,\frac{\pi }{2}]$ area lying between the curve $y=\tan x,y=\cot x$ and $x$-axis is

• A. $\log 2$
• B. $\frac{1}{2}\log 2$
• C. $2\log \left(\frac{1}{\sqrt{2}}\right)$
• D. none of these

Answer 1

The correct option is A $\log 2$

$C_1\to y=tanx,x\in [-\frac{\pi }{3},\frac{\pi }{3}]$

$C_2\to y=x\in [\frac{\pi }{6},\frac{\pi }{6}]$

The region is symmetric about lines $x=\pi /4$ and its first portion is bounded between the lines $x=0$ and $x=\frac{\pi }{4}$

Required Area = $2{\int }_0^{\pi /4}\tan xdy$

$=2\left[\log \right(\sec ){]}_0^{\pi /y}=2[\log \left(\sec \pi /4\right)-\log \left(\sec 0\right)]$

$=2\log \sqrt{2}=\log 2$ sq unit

Area $=\log 2$ Sq. unit