The area of the region bounded by the curve $c: y= an x$, tangent drawn to $c$ at $x=\frac{\pi}{4}$

Question

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

Professor PC Thomas Classes · Answer

$\frac{d y}{d x}\left(ight.$ at $\left.x=\frac{\pi}{4}ight)=2$,\ $P$ is $\left(\frac{\pi}{4}, 1ight), T=\frac{\pi}{4}-\frac{1}{2}$\ Equation of tangent at $P$ is given by\ $y-1=2\left(x-\frac{\pi}{4}ight)$\ Desired area $=$ Area $O P N O-$ Area $	riangle P T N$\ $=\int_{0}^{\frac{\pi}{4}} 	an x \cdot d x-\frac{1}{4}$\ $=(\log \sqrt{2}-0)-\frac{1}{4}=\log \sqrt{2}-\frac{1}{4}$\