<s>[INST] 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?
 [/INST] $$\int \sqrt{x} \text{  }dx=\int 2u^2 \text{  }du=\frac{2}{3}u^3+C.$$
Remember, after substituting, we need to calculate the integral with respect to $u$.
Once that's done, then "back"-substitute $\sqrt x = u$ to get $$\frac 23(x^{1/2})^3 + C = \frac 23 x^{3/2} + C$$
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