Why is the mean x-velocity zero in Maxwell-Boltzmann distribution?

[AStaunton]

I have posted a few questions on maxwell Boltzmann distribution, the problem this time is show:

$$\langle v_{x}\rangle=0$$

I believe the modified maxwell-boltz distrib. for one dimensional case is:

$$f(v_{x})=\sqrt{\frac{m}{2\pi kT}}e^{-\frac{mv_{x}^{2}}{2kt}}$$

My thinking was that simple plug it into the formula:

$$\langle v\rangle=\int_{0}^{\infty}vf(v)dv$$

and the integral should evaluate to 0 but the above clearly doesn't evaluate to 0.

Any advice on how to approach this problem is appreciated.

 

[Redbelly98]

The problem is not dealing with v but instead with v_x, which can be negative. The integration limits should be from -∞ to +∞, not 0 to +∞.