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    Polynomial Function

    Plot

    Alternate form

    x (x^2 - 5) + 1

    Roots

    x≈-2.3301

    x≈0.20164

    x≈2.1284

    Polynomial discriminant

    Δ = 473

    Properties as a real function

    R (all real numbers)

    R (all real numbers)

    surjective onto R

    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 minimum

    min{1 - 5 x + x^3} = 1 - (10 sqrt(5/3))/3 at x = sqrt(5/3)

    Local maximum

    max{1 - 5 x + x^3} = 1 + (10 sqrt(5/3))/3 at x = -sqrt(5/3)

    Definite integral area below the axis between the smallest and largest real roots

    integral_(root of 1 - 5 x + x^3 near x = -2.33006)^(root of 1 - 5 x + x^3 near x = 2.12842) (1 - 5 x + x^3) θ(-1 + 5 x - x^3) dx = -((-1)^(1/4) sqrt(-7160 i + (1602050 i)/(1/2 (1423408309 + 4581951 i sqrt(1419)))^(1/3) - (1602050 sqrt(3))/(1/2 (1423408309 + 4581951 i sqrt(1419)))^(1/3) + i 2^(2/3) (1423408309 + 4581951 i sqrt(1419))^(1/3) + 2^(2/3) sqrt(3) (1423408309 + 4581951 i sqrt(1419))^(1/3)))/(8 sqrt(3))≈-4.1668

    Definite integral area above the axis between the smallest and largest real roots

    integral_(root of 1 - 5 x + x^3 near x = -2.33006)^(root of 1 - 5 x + x^3 near x = 2.12842) (1 - 5 x + x^3) θ(1 - 5 x + x^3) dx = 1/4 sqrt(1/6 (3580 + 1602050/(1/2 (1423408309 + 4581951 i sqrt(1419)))^(1/3) + 2^(2/3) (1423408309 + 4581951 i sqrt(1419))^(1/3)))≈8.63442

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