Question

If $3\int {\sec }^{4}xdx=f\left(x\right)+2\tan x,$ then the value of $f\left(\frac{\pi }{4}\right)$ is equal to
(Assume integration constant to be zero)

Answer 1

$\int {\sec }^{4}xdx=\frac{{\sec }^{2}x\tan x}{3}+\frac{2}{3}\int {\sec }^{2}xdx=\frac{{\sec }^{2}x\tan x}{3}+\frac{2}{3}\tan x+C$
$\therefore 3\int {\sec }^{4}xdx={\sec }^{2}x\tan x+2\tan x$
$(\because C=0)$
So, $f\left(x\right)={\sec }^{2}x\tan x$
$\therefore f\left(\frac{\pi }{4}\right)={\sec }^2\left(\frac{\pi }{4}\right)\cdot \tan \left(\frac{\pi }{4}\right)=2$