e^(-ax)の広義積分[0,∞]

$e^{-ax}$の広義積分$[0,\mathrm{\infty }]$

$$\begin{array}{rcl}{\int }_0^{\mathrm{\infty }}{e}^{-ax}\mathrm{d}x& =& \underset{n\to \mathrm{\infty }}{lim}{\int }_0^n{e}^{-ax}\mathrm{d}x\\ & =& \underset{n\to \mathrm{\infty }}{lim}{\int }_0^{-\mathrm{\infty }}{e}^u\left(\frac{-1}{a}\right)\mathrm{d}u\\ & & \dots u=-ax,\frac{\mathrm{d}u}{\mathrm{d}x}=-a,\mathrm{d}x=\frac{-1}{a}\mathrm{d}u\\ & & \dots x:0\to n,u:0\to -an\\ & =& \underset{n\to \mathrm{\infty }}{lim}\frac{-1}{a}{\int }_0^{-an}{e}^u\mathrm{d}u\\ & =& \underset{n\to \mathrm{\infty }}{lim}\frac{-1}{a}{\left[e^u\right]}_0^{-an}\\ & =& \underset{n\to \mathrm{\infty }}{lim}\frac{-1}{a}[e^{-an}-e^0]\\ & =& \frac{-1}{a}[0-1]\\ & =& \frac{-1}{a}\cdot (-1)\\ & =& \frac{1}{a}\end{array}$$