If f(x)=2^-1x+^-1(2x1+x^2), x1 then f(5) is equal to :

Question

If $f(x)=2	an^{-1}x+\sin^{-1}\left(\dfrac{2x}{1+x^2}ight)$, $x>1$ then $f(5)$ is equal to :

Accepted Answer

For $x>1$, $\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)=\pi-2\tan^{-1}x$, so

$$f(x)=2\tan^{-1}x+\pi-2\tan^{-1}x=\pi\quad\text{for all } x>1.$$

$$f(5)=\pi.$$

Solutions · 1