Integral of x*sinh(x) integration by parts + quick reminder of sinhx coshx derivatives.

00:00 Introduction: we're going to compute the integral of x*sinh(x), where the sinh(x) is the hyperbolic sine function.

00:08 Reminder of sinh(x) definition, cosh(x) definition and sinhx coshx derivatives: we give a quick reminder of the definition of sinh(x) = 1/2*(e^x-e^-x) and definition of cosh(x)=1/2*(e^x+e^-x). Then we work out the derivatives of sinh(x) and cosh(x): it turns out that the derivative of sinhx is coshx and the derivative of coshx is sinhx!

01:20 x*sinh(x) integration by parts: Now we apply integration by parts to integrate x*sinh(x). Let u=x and du=dx; let dv=sinh(x)dx so v=cosh(x) (using the derivative relationship we just figured out). Using the integration by parts formula, we obtain x*cosh(x)-integral(cosh(x))dx, or xcosh(x)-sinh(x)+C, and we're done!