Input
x^3 – x
Polynomial discriminant
Delta = 4
Derivative
d/dx(x^3 – x) = 3 x^2 – 1
Indefinite integral
integral (-x + x^3) dx = x^4/4 – x^2/2 + constant
Local maximum
max{x^3 – x} = 2/(3 sqrt(3)) at x = -1/sqrt(3)
Local minimum
min{x^3 – x} = -2/(3 sqrt(3)) at x = 1/sqrt(3)
Definite integral area below the axis between the smallest and largest real roots
integral_(-1)^1 (-x + x^3) theta(x – x^3) dx = -1/4 = -0.25
Definite integral area above the axis between the smallest and largest real roots
integral_(-1)^1 (-x + x^3) theta(-x + x^3) dx = 1/4 = 0.25