Final answer:
The function xe^-xsin(5x) is annihilated by the given operator due to the properties of an odd function and its integration over the x-axis.
Step-by-step explanation:
The function xe-xsin(5x) is annihilated by the operator. The function in the integrand is odd since it is the product of an odd function (x) and an even function (e-2x/x0). The integral vanishes because the total area of the function about the x-axis cancels the (negative) area below it. As a result, the function is annihilated by the operator.