e^(-ax)の広義積分[0,∞]

\(e^{-ax}\)の広義積分\([0,\infty]\)

\begin{eqnarray} \int_0^\infty e^{-ax} \mathrm{d}x &=&\lim_{n\rightarrow\infty} \int_0^n e^{-ax} \mathrm{d}x \\&=&\lim_{n\rightarrow\infty} \int_0^{-\infty} e^{u} \left(\frac{-1}{a}\right)\mathrm{d}u \\&&\;\ldots\;u=-ax,\;\frac{\mathrm{d}u}{\mathrm{d}x}=-a,\;\mathrm{d}x=\frac{-1}{a}\mathrm{d}u \\&&\;\ldots\;x:0\rightarrow n,\;u:0\rightarrow -an \\&=&\lim_{n\rightarrow\infty} \frac{-1}{a}\int_0^{-an} e^{u}\mathrm{d}u \\&=&\lim_{n\rightarrow\infty} \frac{-1}{a}\left[e^{u} \right]_0^{-an} \\&=&\lim_{n\rightarrow\infty} \frac{-1}{a}\left[e^{-an} - e^{0}\right] \\&=&\frac{-1}{a}\left[0 - 1\right] \\&=&\frac{-1}{a}\cdot(-1) \\&=&\frac{1}{a} \end{eqnarray}