Answer to Question #152420 in Real Analysis for xyz

Question

find the laplace transform of (e^-at -e^-bt)/t

Answer 1

$\begin{aligned} \mathcal{L}\left(\frac{e^{-at} - e^{-bt}}{t}\right) &= \int_s^\infty \mathcal{L}\left(e^{-at} - e^{-bt}\right)\{s\}\,\mathrm{d}s \\&= \int_s^\infty \left(\frac{1}{s + a} - \frac{1}{s + b}\right)\,\mathrm{d}s \\&= \ln\left(\frac{s + a}{s + b}\right) \vert_s^\infty \\&= \lim_{s \to \infty} \ln\left(\frac{1 + \frac{a}{s}}{1 + \frac{b}{s}}\right) - \ln\left(\frac{s + a}{s + b}\right) \\&= \ln(1) - \ln\left(\frac{s + a}{s + b}\right) = -\ln\left(\frac{s + a}{s + b}\right) \\&= \ln\left(\frac{s + b}{s + a}\right) \end{aligned}$