Integrating V(x)DV(X) in NMOS Transistors: A Mathematical Analysis

[xcvxcvvc]

I'm in an electronics course, and the book derives an equation for the current-voltage characteristics of an NMOS transistor. In doing so, it integrate this:
$$\int_{0}^{V_{DS}}V(X)\, dV(X)=\frac{V_{DS}^2}{2}$$I can see that integrating a function F(X) with respect to F(X) turns out to be the same as integrating a single variable such as x with respect to x, but is that mathematically kosher? Can someone convince me that it is?

 

[LCKurtz]

To see that the antiderivative is

$$\int V(x)dV(x) = \frac{V^2(x)} 2+ C$$

Just note that

$$\int V(x)dV(x) = \int V(x)V'(x)dx$$

and that integrand is exactly what you get if you differentiate

$$\frac{V^2(x)} 2+ C$$