Answer
$\int \frac{(1 + \sqrt{x})^{\frac{1}{3}}}{\sqrt{x}} dx = \frac{3}{2}(1+\sqrt{x})^\frac{4}{3} + C$
Work Step by Step
$\int \frac{(1 + \sqrt{x})^{\frac{1}{3}}}{\sqrt{x}} dx\space$ and $\space u = 1 + \sqrt{x}$
$du = \frac{1}{2\sqrt{x}}dx$
Doing the substitution $\space u = 1 + \sqrt{x}$
$\int (u)^{\frac{1}{3}} 2du\space$ => $\space2\int u^\frac{1}{3} du$
Applying the power rule for integrals
$2\int u^\frac{1}{3} du = \frac{3}{2}u^\frac{4}{3} + C$
Backing to x
$\int \frac{(1 + \sqrt{x})^{\frac{1}{3}}}{\sqrt{x}} dx = \frac{3}{2}(1+\sqrt{x})^\frac{4}{3} + C$
