What is the integral from-1 to 1 of x(5-x)^2 ?

The integral from-1 to 1 of x(5-x)^2 is -20/3

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integral from-1 to 1 of x(5-x)^2

\int_{- 1}^{1} x (5 - x)^{2} d x

- \frac{20}{3}

Decimal Notation

- 6.66666 \dots

Solution steps

\int_{- 1}^{1} x (5 - x)^{2} d x

Expand

x (5 - x)^{2} : 25 x - 10 x^{2} + x^{3}

= \int_{- 1}^{1} 25 x - 10 x^{2} + 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

= \int_{- 1}^{1} 25 x d x - \int_{- 1}^{1} 10 x^{2} d x + \int_{- 1}^{1} x^{3} d x

\int_{- 1}^{1} 25 x d x = 0

\int_{- 1}^{1} 10 x^{2} d x = \frac{20}{3}

\int_{- 1}^{1} x^{3} d x = 0

= 0 - \frac{20}{3} + 0

Simplify

= - \frac{20}{3}

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What is the integral from-1 to 1 of x(5-x)^2 ?
The integral from-1 to 1 of x(5-x)^2 is -20/3