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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:     JEE Main Online Paper (Held On 19 April 2016)

A) \[\frac{1}{2}\left( \log 2-\frac{1}{2} \right)\]

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

C) \[\frac{1}{2}\left( 1-\log 2 \right)\]                          

D) \[\frac{1}{2}\left( 1+\log 2 \right)\]

Correct Answer: A

Solution :

The given curve is y = tanx                           ...(1) when \[y-1=\left( {{\sec }^{2}}\frac{\pi }{4} \right)\left( x-\frac{\pi }{4} \right)\] or\[y=2x+1-\frac{\pi }{2}\]                                           ?(2) Area of shaded region = area of OPMO-ar\[(\Delta PLM)\] \[=\int_{0}^{\frac{\pi }{4}}{\tan xdx-\frac{1}{2}(OM-OL)PM}\] \[=[\log \sec x]_{0}^{\frac{\pi }{4}}-\frac{1}{2}\left\{ \frac{\pi }{4}-\frac{\pi -2}{4} \right\}\times 1\] \[=\frac{1}{2}\left\{ \log 2-\frac{1}{2} \right\}\text{sq}\,\text{unit}\]