question_answer

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

A)  \[\frac{1}{2}(1-\ell n2)\]            

B)  \[\frac{1}{2}\left( \ell n2+\frac{1}{2} \right)\]

C)  \[\frac{1}{2}(1+\ell n2)\]          

D)  \[\frac{1}{2}\left( \ell n2-\frac{1}{2} \right)\]

Correct Answer: D

Solution :

Area \[=\int\limits_{0}^{\frac{\pi }{4}}{(\tan x)dx-\frac{1}{2}\left( \frac{1}{2} \right)\,(1)=\frac{1}{2}\,\left( \ell n2-\frac{1}{2} \right)}\]