Question

If $\int \frac{\sqrt{1-\sqrt{x}}}{\sqrt{x(1+\sqrt{x})}}dx = 2f(x)-2\sin^{-1}\sqrt{x}+c$, then $f(x)=$

• A. $\text{Sech}^{-1}\sqrt{x}$
• B. $\text{Cosec}^{-1}\sqrt{x}$
• C. $\log\left(\frac{1+x}{x}\right)$
• D. $\log\left(\frac{\sqrt{1+x}-1}{\sqrt{x}}\right)$

Hint:

Integrals involving $\sqrt{\frac{a-x}{a+x}}$ are often simplified by the trigonometric substitution $x=a\cos(2\theta)$. In this problem, substituting for the inner function, $t=\sqrt{x}$, followed by trigonometric substitution or rationalization is the standard approach.

Answer 1

The Correct Option is A