Feynman's Trick: Integral from 0 to ∞ [(e^(-x)-e^(-bx))/x] dx

\begin{align*}
& I(b) = \int_0^\infty \dfrac{e^{-x}-e^{-bx}}{x} dx = \ln{(b)}\\& \\&{\rm{Proof}}: \\& \\&I'(b) = \int_0^\infty e^{-bx} dx = \dfrac{1}{b} \\& \\&\therefore I(b) = \ln{(b)} + C \\& \\&\therefore b=1 \Longrightarrow C=0 \\& \\&\therefore I(b) = \int_0^\infty \dfrac{e^{-x}-e^{-bx}}{x} dx = \ln{(b)} \\& \\&{\rm{Example}}: \\& \\&I(e) = \int_0^\infty \dfrac{e^{-x}-e^{-ex}}{x} dx = \ln{(e)} = 1
\end{align*}

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