'Unusual' mathematical step in integration

[jdbbou]

I'm following a derivation in my lecture notes of total average particle number in an ideal classical gas (statistical physics approach). I follow it to the line (though the specific terms don't matter):
$$\left<N\right> = e^{\mu/\tau} \frac{\pi}{2} \int_0^\infty \left(n \,dn \,e^{- \frac{\hbar^2 \pi^2}{2m L^2 \tau} n^2} \right)$$
at which point, the substitution $n \, dn = \frac{1}{2} d(n^2)$ is made. I've never seen a substitution like this before and don't understand how it is a valid mathematical step. It's as if the $n$ alone has been integrated, without concern for any integration by parts, and the differential term has spontaneously changed to $d(n^2)$? I can't figure out how this could result from a u-substitution. I expect I'm missing something obvious.

Can anyone tell me the name of this manipulation, or give me some intuition as to why this step is valid?

 

[DrDu]

Its just a change of variables $n -> y=f(n)=n^2/2$ or $n= \sqrt{2y}$. So $dn=(d2\sqrt{y}/dy) dy= 1/n dy$ or $ndn=dy$.