Final answer:
The antiderivative of the given function f(x) = (8)/(x^2) - (8)/(x^5) is F(x) = -8/(x) - 2/(x^4) + 10, with F(1) = 0.
Step-by-step explanation:
To find the antiderivative of the given function f(x) = (8)/(x^2) - (8)/(x^5), we can break it down into two separate terms. For the first term, (8)/(x^2), the antiderivative is -8/(x). For the second term, (8)/(x^5), the antiderivative is -2/(x^4). Therefore, the antiderivative of f(x) is F(x) = -8/(x) - 2/(x^4).
To find the value of the constant of integration, we can use the given condition F(1) = 0. Plugging in x = 1 into the antiderivative, we get F(1) = -8/(1) - 2/(1^4) = -8 - 2 = -10. Since F(1) = 0, we can set the constant of integration as 10. Therefore, the antiderivative of f(x) with F(1) = 0 is F(x) = -8/(x) - 2/(x^4) + 10.