How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
2026-03-28 20:03:26.1774728206
Special Integral Proof
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I think what you meant is $$\int^{\infty}_{0}x^{2n-1}e^{-ax^2}\,dx=\frac{1}{2}\int^{\infty}_{0}y^{n-1}e^{-ay}\,dy=\frac{1}{2a^n}\int^{\infty}_{0}z^{n-1}e^{-z}\,dz=\frac{\Gamma(n)}{2a^n}$$ where we have used $x^2=y$ and $ay=z$. Otherewise the question does not make sense because for $a=1$ the your integral would diverge while the right side is a finite value.