Approximation to An Exponential Integral

Question

Is there any analytical approximation to an integral of this form? $$ \int_{0}^t e^{-\left(\frac{x}{\lambda}\right)^k} dx $$

This comes from integrating $(1-F(x))$, for $F(x)$ being the CDF of a Weibull distribution.

Answer 1

Let $z=(x/a)^{k}$ \begin{align} \int\limits_{0}^{t} \mathrm{e}^{-(x/a)^{}k} dx &= \frac{a}{k} \int\limits_{0}^{(t/a)^{k}} \mathrm{e}^{-z} z^{-1+1/k} dz \\ &= \frac{a}{k} \gamma (1/k, (t/k)^{k} ) \end{align} $\gamma(s,z) = \int\limits_{0}^{z} t^{s-1} \mathrm{e}^{-t} dt$ is the lower incomplete gamma function.