Question

The area of the region above the x-axis bounded by the curve $y=\tan x,0\le x\le \frac{\pi }{2}$ and the tangent to the curve at $x=\frac{\pi }{4}$ is :

• A. $\frac{1}{2}\left(\log 2-\frac{1}{2}\right)$
• B. $\frac{1}{2}\left(\log 2+\frac{1}{2}\right)$
• C. $\frac{1}{2}(1-\log 2)$
• D. $\frac{1}{2}(1+\log 2)$

Answer 1

Answer: (A)

The given curve is $y=\tan x$ ...(1)
when $x=\frac{\pi }{4},y=1$
Equation of tangent at P is
$y-1=\left({\sec }^2\frac{\pi }{4}\right)\left(x-\frac{\pi }{4}\right)$

or $y=2x+1-\frac{\pi }{2}$ ...(2)
Area of shaded region
= area of OPMO - ar ($\Delta$PLM)
$=\underset{0}{\overset{\frac{\pi }{4}}{\int }}\tan xdx-\frac{1}{2}\left(OM-OL\right)PM$
$={\left[\log \sec x\right]}_0^{\frac{\pi }{4}}-\frac{1}{2}\left\{\frac{\pi }{4}-\frac{\pi -2}{4}\right\}\times 1$
$=\frac{1}{2}\left[\log 2-\frac{1}{2}\right]$ sq unit