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

解

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

- 9

解答ステップ

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

定数を除く:

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

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

拡張

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

= 4 \cdot \int_{- 1}^{2} x - x^{3} 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

= 4 (\int_{- 1}^{2} x d x - \int_{- 1}^{2} x^{3} d x)

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

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

= 4 (\frac{3}{2} - \frac{15}{4})

簡素化

4 (\frac{3}{2} - \frac{15}{4}) : - 9

= - 9