TITLE: How to integrate with respect to $x^2$?
QUESTION [6 upvotes]: $$\int x\,dx^2$$
I'm having trouble comprehending this question. I can perform substitution, but when I do I come out wrong. Here's how I've happened to handle this:
$\int x\,dx^2$
$u=x^2$
$du=2x\,dx$
$dx=\frac {du}{2x}$
$\frac12 \int x\frac{du}{2x}$
$\frac 12\int\frac12\,du$
$\frac u4$
$\frac {x^2}{4}$
It says that it's wrong. Me showing my work was just a desperate attempt at solving it. I'd really like a guide through the problem please. 
Also, apparently the answer is $\frac {2x^3}{3}$

REPLY [2 votes]: you can solve this integration by another method.
As we are familiar with integrating with dx instead of $dx^2$, so it will be easier if we translate it to an integral with dx.
Let $y = \int x\,dx^2$
so $\frac{dy}{dx^2}=x$ then we divide by $dx$
to get $\frac{\frac{dy}{dx}}{\frac{dx^2}{dx}} = x$.
$\frac{dx^2}{dx}=2x$, that mean $\frac{y'}{2x}=x$ so $\frac{dy}{dx}=2x^2$ so $y = \frac{2x^3}{3}+k$