Question

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

• A. $log2-\frac{1}{4}$
• B. $\frac{1}{2}log2-\frac{1}{4}$
• C. $log2-\frac{1}{2}$
• D. None

Answer 1

The correct option is B $\frac{1}{2}log2-\frac{1}{4}$



$y=tanx\Rightarrow \frac{dy}{dx}=se{c}^{2}x$. $Atx=\frac{\pi }{4},\frac{dy}{dx}=2$
Equation of the tangent is $y-1=2(x-\frac{\pi }{4})$
This meets $x-axisat(\frac{\pi }{4}-\frac{1}{2},0)$
Area = Area of OAC - Area of $\Delta$ ABC
$={\int }_0^{\frac{\pi }{4}}tanxdx-\frac{1}{2}BC\times AC=logsecx{]}_0^{\frac{\pi }{4}}-\frac{1}{2}[\frac{\pi }{4}-(\frac{\pi }{4}-\frac{1}{2})]\times 1$
$=log\sqrt{2}-\frac{1}{4}$ sq. unit.