Input
1 – 5 x + x^3
Alternate form
x (x^2 – 5) + 1
Polynomial discriminant
Delta = 473
Derivative
d/dx(1 – 5 x + x^3) = 3 x^2 – 5
Indefinite integral
integral (1 – 5 x + x^3) dx = x^4/4 – (5 x^2)/2 + x + constant
Local maximum
max{1 – 5 x + x^3} = 1 + (10 sqrt(5/3))/3 at x = -sqrt(5/3)
Local minimum
min{1 – 5 x + x^3} = 1 – (10 sqrt(5/3))/3 at x = sqrt(5/3)
Definite integral
integral_(-((1 + i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) – (5 (1 – i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)))^(-((1 – i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) – (5 (1 + i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3))) (1 – 5 x + x^3) dx = -((1 – i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) + ((1 + i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) + (5 (1 – i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)) – (5 (1 + i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)) – 1/4 (-((1 + i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) – (5 (1 – i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)))^4 + 1/4 (-((1 – i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) – (5 (1 + i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)))^4 – 5 (1/2 (-((1 – i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) – (5 (1 + i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)))^2 – 1/2 (-((1 + i sqrt(3)) (1/2 (-9 + i sqrt(1419)))^(1/3))/(2 3^(2/3)) – (5 (1 – i sqrt(3)))/(2^(2/3) (3 (-9 + i sqrt(1419)))^(1/3)))^2)~~-8.63442