Answer
$\frac{3}{2}(1+\sqrt x)^{4/3}+c$
Work Step by Step
$u=1+\sqrt x$, therefore $du=\frac{1}{2 \sqrt x}dx$. Let's rewrite the integral! $\int \frac{(1+\sqrt x)^{1/3}}{\sqrt x}dx$
We do the substitution! $\int \frac{(1+\sqrt x)^{1/3}}{\sqrt x}dx=\int u^{1/3}(2du)=2\int u^{1/3}du=2\frac{u^{4/3}}{4/3}+c=\frac{3}{2}u^{4/3}+c$ We re-do the substitution! $\frac{3}{2}u^{4/3}+c=\frac{3}{2}(1+\sqrt x)^{4/3}+c$
