What is the value of integration over zero to infinity of $exp^{-x^n}$?

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What is general formulation of an integration i.e., $\int_0^\infty e^{-x^n} dx$. Well I did some calculation but the answer is wrong, but am not getting where did I got wrong.please check out the solution I did

But if that's correct, it doesn't work for anything except $n=1$.

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Your error is that $e^{-x^n} =e^{-y^{n/2}} $, not $e^{-y}$.