Seems impossible and so either we have user error (most likely) or Mathematica has an integration bug. Here's the integral:
Integrate[(1 - Exp[(-k)*x^a])/(1 - Exp[-k]), {x, 0, 1}, Assumptions -> {k > 0, a > 0}]
And here's the answer I get.
(E^k*(a*k^a^(-1) - Gamma[a^(-1)] + Gamma[a^(-1), k]))/(a*(-1 + E^k)*k^a^(-1))
Let's evaluate this expression at a=0.1 and k=0.1 We get zero! But this can't be. If we plot the function, we see that it is non-negative on the interval from 0 to 1 and assumes positive values in lots of places. And if I do numerical integration on the original expression, I get 0.912835, which looks about right.
What's going on? Is there a fix? Am I missing something?
By the way, there are other integrals similar to this, which also yield results that do not seem possible. Here's the integral on which I discovered what sure looks like a problem. I believe this integral should never be greater than 1/2. And, yet, when I evaluate it for various values of k and a I get values well over 1/2.
Integrate[(1 - Exp[(-k)*x^a])/(1 - Exp[-k]) - x, {x, 0, 1}, Assumptions -> {k > 0, a > 0}]