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

解

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

1296

解答ステップ

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

定数を除く:

\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

拡張

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

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

総和規則を適用する:

\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})

簡素化

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

= 1296