Evaluate the integral $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} dx$. Which of the following functio

Question

Evaluate the integral $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} dx$. Which of the following functions represent the integral? (where $C, C'$ are arbitrary constants)

Professor PC Thomas Classes · Answer

Substitute $x = \cos^2	heta \implies dx = -2\cos	heta\sin	heta d	heta$. $I = \int 	an(	heta/2) (-2\cos	heta\sin	heta) d	heta = -4 \int \sin^2(	heta/2)\cos	heta d	heta = -2 \int (1-\cos	heta)\cos	heta d	heta = -2\sin	heta + 	heta + \frac{1}{2}\sin 2	heta + C$. Converting back to $x$: $	heta = \cos^{-1}\sqrt{x}$, $\sin	heta = \sqrt{1-x}$. Result is $\cos^{-1}\sqrt{x} - 2\sqrt{1-x} + \sqrt{x-x^2} + C$. Options A, B, and C are algebraic expansions or co-function equivalents of this expression.