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

The integral from 0 to 2 of 3x(x-2) is -4

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

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

- 4

Solution steps

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

Take the constant out:

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

= 3 \cdot \int_{0}^{2} x (x - 2) d x

Expand

x (x - 2) : x^{2} - 2 x

= 3 \cdot \int_{0}^{2} 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

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

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

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

= 3 (\frac{8}{3} - 4)

Simplify

3 (\frac{8}{3} - 4) : - 4

= - 4

What is the integral from 0 to 2 of 3x(x-2) ?
The integral from 0 to 2 of 3x(x-2) is -4