integral from-3 to 1 of x(x-1)(x+3)

解

\int_{- 3}^{1} x (x - 1) (x + 3) d x

\frac{32}{3}

十進法表記

10.66666 \dots

解答ステップ

\int_{- 3}^{1} x (x - 1) (x + 3) d x

拡張

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

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

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

\int_{- 3}^{1} x^{3} d x = - 20

\int_{- 3}^{1} 2 x^{2} d x = \frac{56}{3}

\int_{- 3}^{1} 3 x d x = - 12

= - 20 + \frac{56}{3} - (- 12)

簡素化

- 20 + \frac{56}{3} - (- 12) : \frac{32}{3}

= \frac{32}{3}