1
$\begingroup$

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?

$\endgroup$
6
  • 1
    $\begingroup$ Also, I apologize ahead of time for any confusion caused by my question, as I am just starting AP Calculus BC at school. $\endgroup$
    – Alex K
    Aug 2, 2021 at 3:45
  • 3
    $\begingroup$ "most" integrals aren't doable in closed form, so usually it isn't worth trying a random integral; wolfram can't find a solution for the gamma variant wolframalpha.com/input/… . In the case of a step function that's a sum which also looks very hard to me $\endgroup$ Aug 2, 2021 at 3:56
  • 1
    $\begingroup$ To continue with Calvin's point, the integrals we give you in class are very special cases that can actually be done in a simple analytic closed form. It's like when you first learn polynomials, we only give you ones that can be factored over the integers, even though most can't. $\endgroup$
    – Alan
    Aug 2, 2021 at 4:14
  • 1
    $\begingroup$ Unsurprisingly, wolfram alpha doesn't find a closed form $\endgroup$ Aug 2, 2021 at 4:15
  • 1
    $\begingroup$ @Alan That is so true! We are always taught about the "nice" subset of objects, but usually the vast majority are pathological. Some examples of this: (1) Almost all functions are nowhere continuous. (2) Almost all continuous functions are nowhere differentiable. (3) Almost all functions that obey linearity, $f(x+y) = f(x)+f(y)$, are nowhere continuous. (4) Almost all real numbers are incomputable (this one is the most troubling, to me). $\endgroup$ Aug 2, 2021 at 4:26

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.