Can anyone help with the verification that the final integral over infinity = 1?

Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS” (J. Solar Energy Engr., 1982: 107-111) proposes the Rayleigh distribution, with pdf

f(x;

)=

for x>0 and 0 otherwise as a model for the X distribution.

a) Verify that f(x;

) is a legitimate pdf.

Rule 1: f(x)

. All powers of e are positive everywhere so it is left to show

is positive.

must be positive and x is always greater than or equal to zero. Therefore the function is non-negative.

Rule 2:

let u=

then du=

so

Is

= 1?? Or did I screw up royally? Or both?

The problem states that x is non-negative so the integral should go from 0 to