What is the integral from 0 to 2 of sgn(x-x^3) ?

The integral from 0 to 2 of sgn(x-x^3) is -2sgn

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integral from 0 to 2 of sgn(x-x^3)

\int_{0}^{2} s g n (x - x^{3}) d x

- 2 s g n

Solution steps

\int_{0}^{2} s g n (x - x^{3}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= s g n \cdot \int_{0}^{2} x - x^{3} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= s g n (\int_{0}^{2} x d x - \int_{0}^{2} x^{3} d x)

\int_{0}^{2} x d x = 2

\int_{0}^{2} x^{3} d x = 4

= s g n (2 - 4)

Simplify

= - 2 s g n

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What is the integral from 0 to 2 of sgn(x-x^3) ?
The integral from 0 to 2 of sgn(x-x^3) is -2sgn