Question

\[\int\limits_{0}^{\infty} x ^{-4} dx = 1/4! \]
prove it by gamma function.

Answer 1

This is false... The integral you have there does not converge, it grows without bound as the upper bound approaches infinity. The other number you have there that the integral is equal to is the constant 1/24, 4!=24. Is this not what you meant to ask?