The Super Gaussian Integral (The Art of Integration)

You know about the Gaussian integral, which involves an exponential with -x^2 inside. But what if the power was another positive even integer, like 4, 6, or 8? I call this generalized integral the Super Gaussian integral. A nice, closed-form might seem impossible but with a simple substitution we can transform this into a gamma function value. This is basically a nice application of the gamma function. You'll want to be familiar with two results, the Gaussian integral (n = 1) and the gamma function. Videos are linked below.
The Gaussian Integral: • The Gaussian Integral
The Gamma Function: • The Gamma Function
The Art of Integration is an ongoing series where we evaluate integrals with techniques that are not typically taught in the calculus sequence. This is a great way for students in science, engineering, and mathematics to strengthen their integration skills and creativity in solving problems. Most of the problems should be accessible to students that have covered the integration methods from calculus 2.
Looking for a specific problem or topic? Try checking my website:
www.blacktshir...
► Artist Attribution
Music By: "After The Fall"
Track Name: "Pieces"
Music Published by: Chill Out Records LLC
License: Creative Commons Attribution 4.0 International (CC BY - 4.0)
Full license here: creativecommon...

[@wollyben]

Thanks man, this is awesome, this helped me to evaluate a related integral (x^(2n) e^(-x^4) dx).

[@manojsurya1005]

Good video 👍,I left the ans at (1/n)Γ(1/2n),I did not simplify it using the zΓ(z) formula

[@BlackTshirtMathProfessor]

That’s okay, it’s not necessary. I only did it because it makes taking a limit as n approaches infinity easier. In this the super Gaussian integral approaches 2.

[@abhishekbhamare4002]

Nice 👍

[@BlackTshirtMathProfessor]

Thanks!

[@maibisarshad7453]

Please make a video on complex number

[@BlackTshirtMathProfessor]

It’s on my list!