Evaluate $\int {e^{(x!)}}^2dx$

Question

How would I evaluate this integral? $$\int {e^{(x!)}}^2dx$$ Obviously elementary functions will not work. The structure of the integrand looks similar to erfi, but there is an $x!$. As someone pointed out in one of my previous questions, the factorial is only defined on integers, so this integral wouldn't be integrable for real numbers. However, if you used a gamma function, would it be possible to integrate this? And if I didn't use a gamma function and instead considered $x!$ as a step function over positive integers, would it be possible to integrate this?

Answer 1

This form of integral will almost certainly not be able to be expressed in terms of 'elementary' functions (i.e. polynomials, trig functions, exponential functions, and logarithms). The gamma function has singularities at the negative integers so it would be important to chose your bounds carefully.

Otherwise, numerical integration would likely be the best option. The simplest algorithms are the trapezoidal rule or Simpson's rule, but these will require a very fine mesh for large values of the upper bound. More advanced algorithms are likely to exist which can be used to evaluate the integral more efficiently, but finding the appropriate algorithm means searching the literature for something suitable.

Also, software such as Mathematica, Matlab, Maple, etc. have built in algorithms to perform numerical integration.