Weibull function

I need to solve for t the integral of the 3 parameters Weibull function, other characters being constants. See the formula herewith. Can anyone help me on this? Thank you.

http://pwp.netcabo.pt/produtec/varios/formula.jpg

What is R(t)? How about the gamma symbol, does it represent the gamma function?

I'm not sure I quite understand what you want: calculate t given f(t)? Calculate the integeral given a value of alpha?...?

The integral can't be solved by analytic methods unless 1+1/alpha is an integer. If you want to use numerical methods and given that Gamma(x) = (x-1)! you may want to take a look at this thread
Also here's some stuff about the Weibull function in case it may be of interest to you.