Consider the function f ( x ) = 8/x^2 − 2/x^6. Let F ( x ) be the antiderivative of f ( x ) with F ( 1 ) = 0. F(x) equals......

Answer 1

F(x) is the antiderivative of f(x), can be written this way:

∫fx)dx = F(x) + c

where c is constant.

Fx) = ∫(8/x² - 2/x⁶) dx = ∫(8x^-2 - 2x^-6) dx

F(x) = -8x^-1 + (2/5)x^-5 + c

Substitute for F(1) = 0. Meaning when x=1, y=0

F(1) = -8(1)^-1 + (2/5)(1)^-5 + c = 0

-8 + 2/5 + c = 0

-38/5 + c = 0

c = 38/5

Therefore:

F(x) = -8x^-1 + (2/5)x^-5 + 38/5

or

F(x) = -8/x +2/(5x⁵) + 38/5