omgg thank you so much!!!

Your pdf is the opposite of the derivative of the cdf. Also your cdf as per the research

1 - E^(-\[Lambda] ((x/(-1 + x))^\[Alpha]) )

has some problems: it is complex-valued when alpha is noninteger. It is negative, decreasing and unbounded when alpha is an odd integer. It is only plausible when alpha is an even integer.

S = Integrate[(\[Alpha] \[Lambda])/(x - 1)^2 (x/(x - 1))^(\[Alpha] - 
    1)  Exp[-\[Lambda]  (x/(x - 1))^\[Alpha]], {x, 0, t}, 
  Assumptions -> {0 < t <= 1, \[Alpha] > 0, \[Lambda] > 0}]

(*Mathematica Can compute the CDF*)

(* -1 + E^(-E^(I \[Pi] \[Alpha]) (-(t/(-1 + t)))^\[Alpha] \[Lambda]) *)

(*Is not the same as: 1-\[ExponentialE]^(-\[Lambda]((t/(-1+t))^\[Alpha]) ) what you say ? *)

1 - E^(-\[Lambda] ((x/(-1 + x))^\[Alpha]) ) /. x -> 1/2 /. \[Alpha] ->1/3 /. \[Lambda] -> 2 // N
(*1.05907 + 0.363107 I*)

 (* Can there be complex numbers ?*)

Regards M.I.

here is the notebook ,will you please give it a look? :) the function i am working with is the pdf of the kies distribution ...i got this form of the pdf from a past research

Would you edit the code so that it can be pasted into a notebook? Alternatively, attaching a notebook would be helpful. What you've given is missing a few backslashes. For example: [Alpha] instead of \[Alpha].