Wir betrachten die Substitution

${\displaystyle {}y=bx\,}$

bzw.

${\displaystyle {}x={\frac {1}{b}}y\,}$

Aus den Grenzen ${\displaystyle {}1}$ und ${\displaystyle {}a}$ werden dabei die Grenzen ${\displaystyle {}b}$ und ${\displaystyle {}ab}$ und es gilt

${\displaystyle {}\int _{1}^{a}{\frac {1}{x}}dx=\int _{b}^{ab}{\frac {b}{y}}d{\frac {1}{b}}y=\int _{b}^{ab}{\frac {1}{y}}dy\,.}$

Somit ist

${\displaystyle {}{\begin{aligned}\int _{1}^{ab}{\frac {1}{x}}dx&=\int _{b}^{ab}{\frac {1}{x}}dx+\int _{1}^{b}{\frac {1}{x}}dx\\&=\int _{1}^{a}{\frac {1}{x}}dx+\int _{1}^{b}{\frac {1}{x}}dx.\end{aligned}}}$