I remember the following integral identity but not sure of the name. I tried to search all the names I remember (like beta integrals, gamma integrals, etc.) but couldn't find for the one given below.
If someone knows the name, please leave an answer.
$$\int_0^{\infty } x^n e^{-b x^m} \, dx = \frac{\Gamma(\frac{n+1}{m})}{mb^{\frac{n+1}{m}}} $$
First of all we have to restrict the value of $b$ to $b>0$ and the value of $n$ as $n>-1$, otherwise the integral would not converge at all. Now enforcing the substitution $bx^m\mapsto x$ we obtain the following
\begin{align*} \int_0^\infty x^ne^{-bx^m}\mathrm dx&=\int_0^\infty \left(\frac xb\right)^{\frac nm}e^{-x}\left[\frac1{b^{\frac1m}}\frac1mx^{\frac1m-1}\right]\mathrm dx\\ &=\frac1{b^{\frac{n+1}m}}\frac1m\int_0^\infty x^{\frac{n+1}m-1}e^{-x}\mathrm dx\\ &=\frac1{b^{\frac{n+1}m}}\frac1m\Gamma\left(\frac{n+1}m\right) \end{align*}
Your given identity is almost right, to be precise only the factor $1/2$ has to be replaced by $1/m$. However, hence this is only a straightforward application of the substitution $bx^m\mapsto x$ I have doubts that there is a separate name for this identity.