Question:medium

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

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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.

Updated On: Mar 30, 2026

$\text{Sech}^{-1}\sqrt{x}$
$\text{Cosec}^{-1}\sqrt{x}$
$\log\left(\frac{1+x}{x}\right)$
$\log\left(\frac{\sqrt{1+x}-1}{\sqrt{x}}\right)$

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The Correct Option is A

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