The value of the CDF is supposed to be one for particle sizes over 50µm but this is not the case. Is there a coefficient to apply to the integration ?
Well, you simply must scale your "probability density" so that it scales to one. Then integrate it the same way you did to get your second plot, and it will be right (having a horizontal asymptote at 1).
Or, as @Nick Cox says in a comment, it is much easier (and equivalent), to just divide the cumulative values with its maximum.