This is the question I have: Find $ \pi! $
To find $\pi!$, I evaluated the gamma function at $\pi+1$ but ended up with a recursive equation as shown below.
$$ \int_{0}^\infty t^{\pi} e^{-t} dt = \pi \int_{0}^\infty t^{\pi-1} e^{-t} dt $$
I tried using integration by parts. I know that the answer is around $ 7.188 $ using online calculators but am unable to find the workings.
What is the method used to find $\pi!$ ? Thank you for your help.
Integrals can be approximated numerically, without requiring a prior symbolic solution. You only need to be able to evaluate the integrand. This is typically how the online calculators produce their numerical approximations.
For example, Simpson's rule says that the value of an integral can be approximated by: $$\int_a^b f(x)\; dx \approx \frac{b-a}6 \left(f(a) + 4f\biggl(\frac{a+b}2\biggr)+f(b)\right)$$
with the error depending on how much $f(x)$ is like a parabola on the interval $[a,b]$. By dividing the interval of interest into sub-intervals, approximating the value of the integral on each sub-interval, and summing the results, one can obtain good approximations even to functions that are not parabolas.