∫1√(1+x)+√(x)dx = ∫√(1+x) √(x)(√(1+x)+√(x)) (√(1+x) √(x))dx = ∫ nbsp; nbsp;(√(1+x) √(x))dx nbsp; = 23(1+x)^32 23(x)^32 +C ...

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Antiderivative

Evaluate: ∫1√...

Question

Evaluate: $\int \frac{1}{\sqrt{1 + x} + \sqrt{x}} d x$

\frac{2}{3} (1 + x)^{\frac{3}{2}} - \frac{2}{3} (x)^{\frac{3}{2}} + C

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\frac{4}{3} (1 + x)^{\frac{3}{2}} - \frac{2}{3} (x)^{\frac{3}{2}}

+C$

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\frac{3}{2} (1 + x)^{\frac{3}{2}} - \frac{2}{3} (x)^{\frac{3}{2}}

+ C

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\frac{4}{3} (1 + x)^{\frac{3}{2}} - \frac{4}{3} (x)^{\frac{3}{2}}

+C$

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Solution

The correct option is A $\frac{2}{3} (1 + x)^{\frac{3}{2}} - \frac{2}{3} (x)^{\frac{3}{2}} + C$
$\int \frac{1}{\sqrt{1 + x} + \sqrt{x}} d x$

$= \int \frac{\sqrt{1 + x} - \sqrt{x}}{(\sqrt{1 + x} + \sqrt{x}) (\sqrt{1 + x} - \sqrt{x})} d x$

$= \int (\sqrt{1 + x} - \sqrt{x}) d x$

$= \frac{2}{3} (1 + x)^{\frac{3}{2}} - \frac{2}{3} (x)^{\frac{3}{2}}$ $+ C$

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