# How do you find the most general antiderivative of the function f(x) = x(8 - x)^2?

Apr 11, 2015

$f \left(x\right) = x {\left(8 - x\right)}^{2}$ multiply it out to get standard form for a polynomial, then antidifferentiate term by term.

$f \left(x\right) = x {\left(8 - x\right)}^{2} = x \left(64 - 16 x + {x}^{2}\right) = {x}^{3} - 16 {x}^{2} + 64 x$

Whose antiderivative is: $\frac{1}{4} {x}^{4} - \frac{16}{3} {x}^{3} + 32 {x}^{2} + C$