Question

If $g^{\prime }\left(\tan x\right)={\sec }^{2}x\forall x\in R-\left\{\frac{(2n+1)\pi }{2}\right\},n\in Z,g\left(2\right)=2.$ Then the value of $g\left(5\right)$ is

Answer 1

Let $\tan x=t$
$\Rightarrow g^{\prime }\left(t\right)=(1+t^2)$
$(\because 1+{\tan }^{2}x={\sec }^{2}x)$
$\Rightarrow \int g^{\prime }\left(t\right)dt=\int (1+t^2)dt$
$\Rightarrow g\left(t\right)=t+\frac{t^3}{3}+C$
Given $g\left(2\right)=2$
$\therefore C=\frac{-8}{3}$
$\Rightarrow g\left(x\right)=x+\frac{x^3}{3}-\frac{8}{3}$
$\therefore g\left(5\right)=5+\frac{125-8}{3}=44$