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

The integral from-2 to 1 of x(2-x^2-x) is -9/4

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

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

- \frac{9}{4}

Decimal Notation

- 2.25

Solution steps

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

Expand

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

= \int_{- 2}^{1} 2 x - x^{3} - x^{2} 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_{- 2}^{1} 2 x d x - \int_{- 2}^{1} x^{3} d x - \int_{- 2}^{1} x^{2} d x

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

\int_{- 2}^{1} x^{3} d x = - \frac{15}{4}

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

= - 3 - (- \frac{15}{4}) - 3

Simplify

- 3 - (- \frac{15}{4}) - 3 : - \frac{9}{4}

= - \frac{9}{4}

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What is the integral from-2 to 1 of x(2-x^2-x) ?
The integral from-2 to 1 of x(2-x^2-x) is -9/4