question_answer

Area of the region bounded by the curve \[y=\tan x,\] tangent drawn to the curve at \[x=\frac{\pi }{4}\] and the x-axis is       [DCE 2001]

A)            \[\frac{1}{4}\]                      

B)            \[\frac{4}{3}\]

C)            \[\log \sqrt{2}-\frac{1}{4}\]   

D)            None of these

Correct Answer: D

Solution :

Shaded area\[=\int_{0}^{\pi /2}{\,\tan x\,dx}\]\[=[\log \sec x]_{0}^{\pi /4}\]            \[=\log \sec \,(\pi /4)-\log \sec 0\]\[=\log \sqrt{2}-\log 1\]\[=\log \sqrt{2}\].