Question

Evaluate $\int \frac{1}{\sqrt{x+3}-\sqrt{x}}dx$

• A. $\frac{2}{3}{(x+2)}^{3/2}+\frac{2}{3}{x}^{3/2}$
• B. $\frac{2}{9}{(x+3)}^{3/2}+\frac{2}{9}x$
• C. $\frac{2}{9}{(x+3)}^{3/2}+\frac{2}{9}{x}^{3/2}$
• D. $\frac{2}{9}{(x+2)}^{1/2}+\frac{2}{9}{x}^{1/2}$

Answer 1

The correct option is C $\frac{2}{9}{(x+3)}^{3/2}+\frac{2}{9}{x}^{3/2}$

Given,

$\int \frac{1}{\sqrt{x+3}-\sqrt{x}}dx$

$=\int \frac{\sqrt{x+3}+\sqrt{x}}{{\left(\sqrt{x+3}\right)}^2-{\left(\sqrt{x}\right)}^2}dx$ ............. Rationalisation

$\int \frac{1}{3}(\sqrt{x}+\sqrt{x+3})dx$

$=\frac{1}{3}(\int \sqrt{x}dx+\int \sqrt{x+3}dx)$

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

$=\frac{2}{9}{(x+3)}^{\frac{3}{2}}+\frac{2}{9}{x}^{\frac{3}{2}}$