Definite integral x*sinh(x), integration by parts with hyperbolic sine, let u=x.

We compute the definite integral x*sinh(x) by using integration by parts. To use integration by parts with hyperbolic sine, let u=x because the derivative of u becomes simpler and du=dx. With dv=sinh(x)dx, we guess the antiderivative, and v=cosh(x). We apply the integration by parts formula, and we are left with one more integral: cosh(x). After getting the antiderivative of cosh(x), we evaluate across the limits of integration.

We get a reminder of the definition of sinh(x) and cosh(x) in terms of exponential functions, and we evaluate the expanded form of the hyperbolic trig functions across the limits of integration. After simplifying, we arrive at the very simple answer, 1/e.