consider the function f(x)=(10/x^2)-(5/x^6). Let F(x) be the antiderivative of f(x) with F(1) = 0. What does F(3) equal?

Answer 1

Let F(x) be the antiderivative of f(x):

F(x)= ∫f(x) dx

F(x) = ∫(10/x²) - (5/x⁶) dx

F(x) = -10/x +(1/x⁵) + c

With F(1) =0.

F(1) = 0 = -10/1 +(1/1⁵) + c

0 = -9 + c

c=9

Therefore:

F(x) = -(10/x) +(1/x⁵) + 9

∴F(3) = -10/3 + 1/3⁵ + 9

F(3) = -10/3 + 1/243 + 9

F(3) = -810/243 + 1/243 + 2187/243

F(3) = 1378/243 ≈ 5.67