Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

Question

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$

As a tenth grader I have no idea what the solutions could be. How would one go about evaluating this without computational engines? I'm asking this here because many complex problems have been tackled here...(eg:Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$).

Any hints or solutions to these integrals would be greatly appreciated.

Note: I don't necessarily want closed forms; special functions are okay. [http://en.wikipedia.org/wiki/List_of_mathematical_functions and http://en.wikipedia.org/wiki/Closed-form_expression]

[[ PS: The graphs for the functions inside the aforementioned integrals are amazing! ]]

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Background:

I was recently in the process of understanding the wonders of the gamma function. It is really fun to attend to derivatives involving subfactorials, factorials and the gamma function. [In case someone is interested, here are some examples of expressions I was solving] :-

$$ \frac{d}{dx} [x!^{!x}!x^{x!}]^{(x!)(!x)} $$ $$ \frac{d}{dx} \frac{\sqrt{1+\arctan(x)}}{\Gamma(x)} $$

The problem arose when I thought of the two aforementioned integrals I have no answer to.

Answer 1

For these kind of whatever integral you can use a telscoping method explained in

https://math.stackexchange.com/a/4186998

Basically you start from the opposite task

$$\int e^{\Gamma(x)} dx =F(x)$$

$$F'(x) = e^{\Gamma(x)}$$

$$F(x)=e^{\Gamma(x)}f(x)$$

This is a general form we expect

$$F'(x)=e^{\Gamma(x)}=e^{\Gamma(x)}f'(x)+e^{\Gamma(x)}\Gamma(x)\Gamma'(x)f(x)$$

This is making

$$f(x)=\frac{1}{\Gamma(x)\Gamma'(x)}$$

the best guess. The rest is simply recursive

$$\int e^{\Gamma(x)} dx = e^{\Gamma(x)} \sum_{n=0}^{+\infty}f_n(x)$$

$$f_0(x)=\frac{1}{\Gamma(x)\Gamma'(x)}$$

$$f_{n+1}(x)=-\frac{f_{n}'(x)}{\Gamma'(x)}$$

Answer 2

These will simply be many small solutions. Here is the graph of the function. Note that I will take the primitive for simplicity.

The first way uses the Maclaurin series for $e^y$. We can integrate term by term:

$$\mathrm{A(a)=\int a^{Γ(x)}dx=\int \sum_{n=0}^\infty \frac{ln^n(a)Γ^n(x)}{n!}dx= \sum_{n=0}^\infty \frac{ln^n(a)}{n!}\int Γ^n(x)dx= \quad\sum_{n=0}^\infty \frac{ln^n(a)}{n!}\int \prod_{n=1}^{x-1} (x-m)^ndx}$$

It would be nice if we could use a truncated version of the Pentagonal Number theorem or we could expand the $(x-m)^n$ into a binomial series. We can also turn the product into a sum via logarithms. I still do not see how to integrate this.

Here is another idea which usually works for integration. Integrating with a Taylor series or using the Riemann Sum definition:

$$\mathrm{\int_a^b y^{Γ(x)}dx=\lim_{n\to\infty}\frac{b-a}{n}\sum_{k=0}^n y^{Γ\left(a+k\frac{b-a}{n}\right)}=\sum_{n=0}^\infty \frac{\frac{d^n}{dx^n}\left(y^{Γ(x)}\right )\big|_{x=r}(x-r)^n}{n!}}$$

I also had the idea to expand $Γ^n(x)$ into factored integral representations of Γ(x) and using the Cauchy repeated integral formula. I tried it, but am unsure if it would be used correctly.

I will add more. Please correct me and give me feedback!