I need to integrate $$\langle |v_x| \rangle = \int^{\infty}_0 |v_x| \sqrt{\frac{m}{2\pi kT}}e^{-mv_x^2/2kT}dv$$ For context this is a Maxwell boltzmann distribution in one dimension, and I've actually been asked to calculate $\langle v_x \rangle$ which is given by $|v_x|f(v)$ where $f(v) = \sqrt{\frac{m}{2\pi kT}}e^{-mv_x^2/2kT}$ is the Maxwell Boltzmann distribution of velocities in the x-direction. Not sure if the question is best put in physics or maths.
I'm a little confused because since $v_x$ is always positive between infinity and zero (I think?) the integral is actually just $\langle v_x \rangle$, since the mod can be ignored if it's always positive. That can't be it though, I've already been asked to calculate $\langle v_x \rangle$ in the first part of the same question. So I suppose my treatment of the mod must be wrong. Any help is appreciated!
Bad physics. It's a velocity distribution and needs to be integrated from $\infty$ to $-\infty$, because velocity can be negative! So then the mod makes a difference to the outcome, the sum has to be split into parts where $v$ is positive and parts where it is negative.