<s>[INST] Using the substitution method for a simple integral I have been playing with the substitution rule in order to test some ideas with computational graphs. One of the things I'm doing is applying the substitution to well known, and easy, integrals. For example, let's use that method to find the indefinite integral for 
$$f(x) = x^2$$
Using the rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, we get 
$$F(x) = \frac{x^3}{3} + C$$
So, let's do the following substitution:
$$u = x^2$$
$$\frac{du}{dx} = 2x \Leftrightarrow dx = \frac{1}{2x} du$$
So, performing the substitution in the integral of $f(x)$ gives us 
$$\int x^2 dx = \int u \frac{1}{2x} du = \frac{1}{2x} \frac{u^2}{2} + C = \frac{(x^2)^2}{4x} + C = \frac{x^3}{4} + C$$
Have I done anything wrong with the substitutions??
Thanks in advance!
 [/INST] $\int \frac{1}{2x}u du$ = $\int \frac{1}{2\sqrt{u}}\cdot u du$ = $ \int \frac{\sqrt{u}}{2}du$ = $u^{\frac{3}{2}}\cdot\frac{2}{3}\cdot \frac{1}{2} + C$
$= \frac{u^{\frac 3 2}}{3} + C = \frac{x^3}{3} + C$ as required.
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