You posted this same question yesterday and have received 16 responses to it. Have you read them? x! is only defined for integers and so does not have a Riemann integral. You could do it as a "Stieljes" integral: [tex]\int x! d\alpha(x)[/tex] where α(x) is the step function. In that case, the integral, from 1 to n, is the sum [tex]\Sigma_1^n x![/tex].