What is the integral from 0 to 2 of x(x-1)(x-2) ?

The integral from 0 to 2 of x(x-1)(x-2) is 0

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integral from 0 to 2 of x(x-1)(x-2)

\int_{0}^{2} x (x - 1) (x - 2) d x

0

Solution steps

\int_{0}^{2} x (x - 1) (x - 2) d x

Expand

x (x - 1) (x - 2) : x^{3} - 3 x^{2} + 2 x

= \int_{0}^{2} x^{3} - 3 x^{2} + 2 x 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

= \int_{0}^{2} x^{3} d x - \int_{0}^{2} 3 x^{2} d x + \int_{0}^{2} 2 x d x

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

\int_{0}^{2} 3 x^{2} d x = 8

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

= 4 - 8 + 4

Simplify

= 0

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What is the integral from 0 to 2 of x(x-1)(x-2) ?
The integral from 0 to 2 of x(x-1)(x-2) is 0