What is the integral from 0 to 3 of x(x-1)(x-3) ?

The integral from 0 to 3 of x(x-1)(x-3) is -9/4

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integral from 0 to 3 of x(x-1)(x-3)

\int_{0}^{3} x (x - 1) (x - 3) d x

- \frac{9}{4}

Decimal

- 2.25

Solution steps

\int_{0}^{3} x (x - 1) (x - 3) d x

Expand

x (x - 1) (x - 3) : x^{3} - 4 x^{2} + 3 x

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

\int_{0}^{3} x^{3} d x = \frac{81}{4}

\int_{0}^{3} 4 x^{2} d x = 36

\int_{0}^{3} 3 x d x = \frac{27}{2}

= \frac{81}{4} - 36 + \frac{27}{2}

Simplify

\frac{81}{4} - 36 + \frac{27}{2} : - \frac{9}{4}

= - \frac{9}{4}

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What is the integral from 0 to 3 of x(x-1)(x-3) ?
The integral from 0 to 3 of x(x-1)(x-3) is -9/4