What is the integral from-1 to 5 of 4x(1-x^2) ?

The integral from-1 to 5 of 4x(1-x^2) is -576

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integral from-1 to 5 of 4x(1-x^2)

\int_{- 1}^{5} 4 x (1 - x^{2}) d x

- 576

Solution steps

\int_{- 1}^{5} 4 x (1 - x^{2}) d x

Take the constant out:

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

= 4 \cdot \int_{- 1}^{5} x (1 - x^{2}) d x

Expand

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

= 4 \cdot \int_{- 1}^{5} x - 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

= 4 (\int_{- 1}^{5} x d x - \int_{- 1}^{5} x^{3} d x)

\int_{- 1}^{5} x d x = 12

\int_{- 1}^{5} x^{3} d x = 156

= 4 (12 - 156)

Simplify

= - 576

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What is the integral from-1 to 5 of 4x(1-x^2) ?
The integral from-1 to 5 of 4x(1-x^2) is -576