Question

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

• A. $\log \sqrt{2}-\frac{\pi }{4}+\frac{{\pi }^2}{16}$ sq. units
• B. $\log \sqrt{2}+\frac{1}{4}$ sq. units
• C. $\log \sqrt{2}$
• D. log 2

Answer 1

The correct option is A $\log \sqrt{2}-\frac{\pi }{4}+\frac{{\pi }^2}{16}$ sq. units

$\stackrel{\pi /4}{\underset{0}{\int }}(\tan x-[2x+(1-\pi /2)\left]\right)dx$
$\log |\sec x|{|}_0^{\pi /4}-x^2{|}_0^{\pi /4}-(1-\pi /2)\left(\pi /4\right)$
$\log |\sqrt{2}|-\frac{{\pi }^2}{16}-\frac{\pi }{4}+\frac{{\pi }^2}{8}$
$=\log \sqrt{2}-\pi /4+\frac{{\pi }^2}{16}$