Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 3

$$\frac{1}{2} \cosh \left(x^{2}+1\right)+C$$

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