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\[(\Delta PLM)\] \[=\int_{0}^{\frac{\pi }{4}}{\tan xdx-\frac{1}{2}(OM-OL)PM}\] \[=[\log \sec x]_{0}^{\frac{\pi }{4}}-\frac{1}{2}\left\{ \frac{\pi }{4}-\frac{\pi -2}{4} \right\}\times 1\] \[=\frac{1}{2}\left\{ \log 2-\frac{1}{2} \right\}\text{sq}\,\text{unit}\]You need to login to perform this action.
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