Integral of square root. We have
$$\int \sqrt{x} \text{  }dx=\int x^{1/2} \text{  }dx=\frac{2}{3}x^{3/2}+C.$$
But, let $u=\sqrt{x}$. Then $x=u^2$, and $dx=2udu$. Substituting, we have
$$\int \sqrt{x} \text{  }dx=\int 2u^2 \text{  }du=\frac{2}{3}x^3+C.$$
Which one is correct, and why?

You have $u= \sqrt{x}$, hence $\int 2u^2 du = \frac{2u^3}{3}+C= \frac{2x^{3/2}}{3}+C$
