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Given the Maxwell-Boltzmann distribution:

f(v) = 4*pi*[m/(2*pi*k*T)]^(3/2)*v^2*exp[(-m*v^2)/(2*k*T)]

Assuming a fixed temperature and mass, one can simplify this equation:

f(v) = a*v^2*exp[-bv^2]

a = 4*pi*[m/(2*pi*k*T)]^(3/2)

b = m/(2*k*T)

In order to calculate the fraction of particles between two speeds v1 and v2, one should evaluate the definite integral:

∫f(v)dv

Can this be done analytically?

Also, this function predicts a non-zero (albeit very small) probability for particles to have a speed greater than the speed of light. Is there a correction to this distribution that takes this into account?