Root Mean Square (RMS) Speed Derived from
Maxwell's Velocity Distribution Function
This is the square root of the mean of the squared velocities and it is most often defined as an integral. The principle use of RMS derives from signal analysis. Clearly it is useless to average a sinusoidal signal that oscillates about zero. Consequently, the average is defined as the mean of the squared signal values, thus never having to add +1 and -1 to get a zero average. After having averaged the squared values, the square root is obtained to get a proper average.
<= Use each velocity2 times the probability of that velocity, thereby losing 1/N.
I am assigning some values so that I can show that the algebra is in fact correct at each step.
Here I have substituted the Maxwell eqn for f(n). And temporarily dropped the final square root. I will put it back at the end.
Here I have moved some constants out of the integration.
Now I am going to do a complex substitution in order to perform the integration. The purpose of the substitutions is to use the integral form

Let
Next I will find an expression for dv
<= this comes from the highlighted v2 equation.
<= and allows a 'dv' expression which still has v in in.
then substituting in v (as solved above) we get
<= Simplified
We will substitute the highlighted equations in 'x' into the velocity distribution.
Just make the substitutions for v2, x2 and dv and show that numerically, it is still giving the same answer.
<= Result of doing the integration symbolically
First step of a simplification.
Second step
Third step
Final form (except recall that we still need to reintroduce the square root.