Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 11

$$\ln |\sinh x|+C$$

Given $$\int \frac{\cosh x}{\sinh x} d x $$ Let $$ u= \sin h x \ \ \ \to du = \cosh x dx$$ Then \begin{aligned} \int \frac{\cosh x}{\sinh x} d x &=\int \frac{1}{u} d u \\ &=\ln |u|+C \\ &=\ln |\sinh x|+C \end{aligned}