Evaluate: 2 x-x+3 d x - Vidyadip

Question

Evaluate:
$\int \frac{2}{\sqrt{x}-\sqrt{x+3}} \cdot d x$

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Answer

$ \int \frac{2}{\sqrt{x}-\sqrt{x+3}} d x$
$=\int \frac{2}{\sqrt{x}-\sqrt{x+3}} \times \frac{\sqrt{x}+\sqrt{x+3}}{\sqrt{x}+\sqrt{x+3}} d x$
$=\int \frac{2(\sqrt{x}+\sqrt{x+3})}{x-(x+3)} d x$
$=-\frac{2}{3} \int(\sqrt{x}+\sqrt{x+3}) d x$
$=-\frac{2}{3} \int x^{\frac{1}{2}} d x-\frac{2}{3} \int(x+3)^{\frac{1}{2}} d x$
$=-\frac{2}{3} \cdot \frac{x^{\frac{3}{2}}}{\left(\frac{3}{2}\right)}-\frac{2}{3} \cdot \frac{(x+3)^{\frac{3}{2}}}{\left(\frac{3}{2}\right)}+c$
$=-\frac{4}{9}\left[x^{\frac{3}{2}}+(x+3)^{\frac{3}{2}}\right]+c .$

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