Final answer:
The constant of integration that makes the integral of sin(3x)cos(5x) equal to zero at x=0 is zero, without needing to perform the integration itself, due to the properties of sine and cosine functions at x=0.
Step-by-step explanation:
To find the value of the constant of integration that makes the integral of sin(3x)cos(5x) equal to zero at x=0, we would typically integrate the function and then solve for the constant, given the initial condition. However, in this case, we can use a property of trigonometric functions: because the integral of a product of a sine and cosine functions over a complete cycle is zero, the constant of integration must be the value that makes the indefinite integral zero when evaluated at x=0.
Without explicitly doing the integral, we can observe that both sin(3x) and cos(5x) are zero at x=0; thus the product is zero, and therefore, the integral evaluated at x=0 will be solely determined by the constant of integration. To make this integral zero at x=0, the constant of integration must also be zero.
To find the value of C, we substitute x = 0 into the integrated expression and set it equal to zero, giving (1/16)(-cos(0) - cos(0)) + C = 0. Simplifying, we have -1/16 + C = 0, so C = 1/16. Therefore, the correct value of the constant of integration is 1/16 (Answer choice D).