Question

\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]

Answer 1

\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}}dx\]

\[Let, 1 + \sqrt{x} = t\]

\[ \Rightarrow \frac{1}{2\sqrt{x}} = \frac{dt}{dx}\]

\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2dt\]

\[Now, \int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}}dx \]

\[ = 2\int t^2 dt\]

\[ = \frac{2}{3} t^3 + C\]

\[ = \frac{2}{3} \left( 1 + \sqrt{x} \right)^3 + C\]