PS  Beta and Theta are numebers, x is the variable.

You may see this paper for your question where a procedure is given.

Conference Paper Preventive Maintenance Scheduling using Weibull Distribution

Dear Marcello, 

your integral ? is, up to a constant, an incomplete gamma function. The attached calculation shows how you can reach it.

Regards, 

Tibor Pogany

Thank you very match because this was very helpful. I’m studying the gamma function because I had never heard before… And I don’t understand the last sentence. Is the mining ϒ(s,a) = Gamma[a, 0, z] in a Mathematica program? But in this case.. what is “z”?

Dear Marcello,

I have permuted the parameters, sorry. So, in TEX speaking: 

\[ \gamma(s, a) = \int_0^a e^{-u} u^{s-1} du \]

is the lower incomplete gamma function, see e.g.

http://en.wikipedia.org/wiki/Incomplete_gamma_function

The Mathematica code for $\gamma(s, a)$ is Gamma[s,0,a], that is

\[ \gamma(s, a) = Gamma[s,0,a] \];

compare 

http://mathworld.wolfram.com/IncompleteGammaFunction.html.

Regards,

TP

Perfect! Thanks a lot!!

Dear Marcello, 

You may see the below links for more detail regarding the optimal time for Preventive maintenance. Hope it will solve ur problem.

Regards

HG

http://link.springer.com/article/10.1007%2Fs13160-012-0099-4

Article Preventive maintenance scheduling of the pulping unit in a paper mill

The simplest calculus of an integral is to approximate it by an infinite sum of of integrals over [t_i, t_i+1] where the function is aproximated by its value at t_i as the length of the interval is small enough.