We can now find
f(x),
which is given by the general antiderivative of
f ′(x) = -2x + 9x2 - 4x3 + 16,
To do so, we refer again to the following antidifferentiation formulas.
A function of the form
g(x) + h(x)
will have the particular antiderivative
G(x) + H(x).
A function of the form
ch(x)
where c is a constant will have the particular antiderivative
cH(x).
A function of the form
xn
where
n != -1
will have a particular antiderivative of the form
xn + 1
n + 1
Applying these formulas to find the general antiderivative of
f ′(x) = -2x + 9x2 - 4x3 + 16
and letting D represent the arbitrary constant gives the following result.