Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 9

$$\ln |\cosh x|+C $$

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