By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} \int_{1}^{x} \frac{a^{t-1}}{\Gamma(t)} \, dt \hspace{5mm} ? \end{align}
It is speculated that the result of the integral will be an infinite series involving either the Bernoulli numbers of the second kind or Stirling numbers and some polynomial related to the Laguerre polynomial.
