What is the integral from-3 to 3 of (9x-x^3)(20x) ?

The integral from-3 to 3 of (9x-x^3)(20x) is 1296

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integral from-3 to 3 of (9x-x^3)(20x)

\int_{- 3}^{3} (9 x - x^{3}) (20 x) d x

1296

Solution steps

\int_{- 3}^{3} (9 x - x^{3}) \cdot 20 x d x

Take the constant out:

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

= 20 \cdot \int_{- 3}^{3} (9 x - x^{3}) x d x

Expand

(9 x - x^{3}) x : 9 x^{2} - x^{4}

= 20 \cdot \int_{- 3}^{3} 9 x^{2} - x^{4} 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

= 20 (\int_{- 3}^{3} 9 x^{2} d x - \int_{- 3}^{3} x^{4} d x)

\int_{- 3}^{3} 9 x^{2} d x = 162

\int_{- 3}^{3} x^{4} d x = \frac{486}{5}

= 20 (162 - \frac{486}{5})

Simplify

20 (162 - \frac{486}{5}) : 1296

= 1296

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What is the integral from-3 to 3 of (9x-x^3)(20x) ?
The integral from-3 to 3 of (9x-x^3)(20x) is 1296