Question

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

• A. $\frac{1}{4}({\log }_{e}4-1)$
• B. $\frac{1}{4}({\log }_{e}2-1)$
• C. $\frac{1}{2}({\log }_{e}4-1)$
• D. $\frac{1}{2}({\log }_{e}2-1)$

Answer 1

The correct option is A $\frac{1}{4}({\log }_{e}4-1)$


Slope of the tangent at $x=\frac{\pi }{4}$ is ${\sec }^2\frac{\pi }{4}=2$
Equation of the tangent at $A$ is,
$y-1=2(x-\frac{\pi }{4})$
$\therefore$ Coordinates of $B$ are $(\frac{\pi }{4}-\frac{1}{2},0)$
$BC=\frac{1}{2}$

Required area
$=$ Area of $OACO-$ Area of $\Delta ABC$
$=\underset{0}{\overset{\pi /4}{\int }}\tan xdx-\frac{1}{2}\times AC\times BC$
$=[{\log }_e\sec x{]}_0^{\pi /4}-\frac{1}{2}\times 1\times \frac{1}{2}=({\log }_e\sqrt{2}-0)-\frac{1}{4}=\frac{1}{4}({\log }_{e}4-1)$