Question

$I={\int }_0^1\sqrt[3]{32x^3-3x^2-x+1}dx$ is

Answer 1

Apply property ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$

$I=\underset{0}{\overset{1}{\int }}\sqrt[3]{32(1-x{)}^3-3(1-x{)}^2-(1-x)+1}dx$
$\Rightarrow I=\underset{0}{\overset{-1}{\int }}\sqrt[3]{32x^3-3x^2-x+1}dx$
$\Rightarrow I=-I$
$\Rightarrow 2I=0$
$\Rightarrow I=0$