Question

Solve: $\underset{0}{\overset{\infty }{\int }}\left(a^{-x}-b^{-x}\right)dx$

Answer 1

$\begin{array}{c}\underset{0}{\overset{\infty }{\int }}\left(a^{-x}-b^{-x}\right)dx\\ =\underset{0}{\overset{\infty }{\int }}{\left(\frac{1}{a}\right)}^{x}dx-\underset{0}{\overset{\infty }{\int }}{\left(\frac{1}{b}\right)}^{x}dx\\ ={\left[{\left(\frac{1}{a}\right)}^x\ln \left(\frac{1}{a}\right)\right]}_0^{\infty }-{\left[{\left(\frac{1}{b}\right)}^x\ln \left(\frac{1}{b}\right)\right]}_0^{\infty }\\ =0-\ln \left(\frac{1}{a}\right)-\left(0-\ln \left(\frac{1}{b}\right)\right)\\ =\ln \left(\frac{1}{b}\right)-\ln \left(\frac{1}{a}\right)=\ln \frac{a}{b}\end{array}$