Consider the function f(x)=((2)/(x^2))-((3)/(x^5)). Let F(x) be the antiderivative of f(x) with F(1) = 0. Find F(3).
To find F(3), we need to calculate the definite integral of f(x) from 1 to 3, since F(x) is the antiderivative of f(x).
First, let's find the indefinite integral of f(x):
∫ f(x) dx = ∫ ((2)/(x^2))-((3)/(x^5)) dx
To evaluate this integral, we can break it down into two separate integrals:
∫ ((2)/(x^2)) dx - ∫ ((3)/(x^5)) dx
Integrating the first term:
∫ ((2)/(x^2)) dx = 2 ∫ (1/(x^2)) dx = 2 * (-1/x) = -2/x
Integrating the second term:
∫ ((3)/(x^5)) dx = 3 ∫ (1/(x^5)) dx = 3 * (-1/(4x^4)) = -3/(4x^4)
Now, let's evaluate the definite integral of f(x) from 1 to 3:
F(3) - F(1) = ∫ f(x) dx from 1 to 3
Substituting the definite integral values:
F(3) - 0 = ∫ f(x) dx from 1 to 3
Using the antiderivative we found earlier:
F(3) = -2/x - 3/(4x^4)
Evaluating F(3) at x = 3:
F(3) = -2/3 - 3/(4(3^4))
= -2/3 - 3/432
= -2/3 - 1/144
= (-288 - 1)/(432)
= -289/432
Therefore, F(3) = -289/432.