Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 4

$$\frac{1}{3} \sinh ^{3} x+C $$

Given $$\int \sinh ^{2} x \cosh x d x$$ Let $$ u=\sinh x \ \ \ \to du =\cosh xdx$$ Then \begin{aligned} \int \sinh ^{2} x \cosh x d x &=\int u^{2} d u \\ &=\frac{u^{3}}{3}+C \\ &=\frac{1}{3} \sinh ^{3} x+C \end{aligned}