Question

Solve:

${\cos }^{-1}\sqrt{\frac{1+x}{2}}$

Answer 1

Consider the following equation.

$y={\cos }^{-1}\sqrt{\frac{x+1}{2}}$

${\cos }^{2}y=\frac{x+1}{2}$

$-2\cos y\sin y\frac{dy}{dx}=\frac{1}{2}$

$-\sin 2y\frac{dy}{dx}=\frac{1}{2}$

$-\underset{}{\overset{}{\int }}\sin 2ydy=\underset{}{\overset{}{\int }}\frac{1}{2}dx$

$\frac{\cos 2y}{2}=\frac{x}{2}+C$

$\cos 2y-x=2C$

Hence, this is the required answer.