x・exp(-α x^2)の広義積分[0,∞]

${\int }_0^{\mathrm{\infty }}x{e}^{-\alpha x^2}\mathrm{d}x$

$$\begin{array}{rcl}& & {\int }_0^{\mathrm{\infty }}x{e}^{-\alpha x^2}\mathrm{d}x\\ & =& {\int }_0^{\mathrm{\infty }}\sqrt{\frac{t}{\alpha }}{e}^{-t}\frac{1}{2\sqrt{\alpha t}}\mathrm{d}t\\ & & \cdots t=\alpha x^2,x=0\to t=0,x=\mathrm{\infty }\to t=\mathrm{\infty },\\ & & \cdots x=\sqrt{\frac{t}{\alpha }}(\mathrm{積}\mathrm{分}\mathrm{範}\mathrm{囲}\mathrm{か}\mathrm{ら}x\ge 0\mathrm{と}\mathrm{す}\mathrm{る}(?))\\ & & \cdots \frac{\mathrm{d}t}{\mathrm{d}x}=2\alpha x,\mathrm{d}x=\frac{1}{2\alpha x}\mathrm{d}t=\frac{1}{2\alpha \sqrt{\frac{t}{\alpha }}}\mathrm{d}t=\frac{1}{2\sqrt{\alpha t}}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{\left(\frac{t}{\alpha }\right)}^{\frac{1}{2}}{e}^{-t}\frac{1}{2\sqrt{\alpha t}}\mathrm{d}t\\ & =& \frac{1}{2{\alpha }^{\frac{1}{2}}\sqrt{\alpha }}{\int }_0^{\mathrm{\infty }}\frac{t^{\frac{1}{2}}}{\sqrt{t}}{e}^{-t}\mathrm{d}t\\ & =& \frac{1}{2\alpha }{\int }_0^{\mathrm{\infty }}{e}^{-t}\mathrm{d}t\\ & =& \frac{1}{2\alpha }{\int }_0^{\mathrm{\infty }}1\cdot e^{-t}\mathrm{d}t\\ & =& \frac{1}{2\alpha }{\int }_0^{\mathrm{\infty }}{t}^0\cdot e^{-t}\mathrm{d}t\\ & =& \frac{1}{2\alpha }{\int }_0^{\mathrm{\infty }}{t}^{1-1}{e}^{-t}\mathrm{d}t\\ & =& \frac{1}{2\alpha }\mathrm{\Gamma }\left(1\right)\cdots \mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}\mathrm{d}t(\mathfrak{Re}(z)>0)\\ & =& \frac{1}{2\alpha }(1-1)!\cdots \mathrm{\Gamma }(n)=(n-1)!(n\mathrm{は}\mathrm{自}\mathrm{然}\mathrm{数})\\ & =& \frac{1}{2\alpha }\cdot 1\cdots 0!=1\\ & =& \frac{1}{2\alpha }\end{array}$$