What is the most general antiderivative of the following function and how do you get it?

Question

g(x)=((5-(4x^2)+(2x^6))/x^6)

Answer 1

Well, this is just 5/x^6 + 4/x^4 + 1, and the antiderivative is gotten by multiplying by x and dividing by one more than the current power of x.
Therefore, it is -1/x^5 - 4/(3 x^3) + x + C, for a constant C.

Answer 2

g(x) = (5 - 4x^2 + 2x^6) / x^6
= 5x^(-6) - 4x^(-4) + 2
So the antiderivative G(x) is given by
G(x) = 5x^(-5) / (-5) - 4x^(-3) / (-3) + 2x + c
= -1/x^5 + 4/(3x^3) + 2x + c.