integral from-3 to 2 of x(x+3)(x-2)

解

\int_{- 3}^{2} x (x + 3) (x - 2) d x

\frac{125}{12}

十進法表記

10.41666 \dots

解答ステップ

\int_{- 3}^{2} x (x + 3) (x - 2) d x

拡張

x (x + 3) (x - 2) : x^{3} + x^{2} - 6 x

= \int_{- 3}^{2} x^{3} + x^{2} - 6 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}^{2} x^{3} d x + \int_{- 3}^{2} x^{2} d x - \int_{- 3}^{2} 6 x d x

\int_{- 3}^{2} x^{3} d x = - \frac{65}{4}

\int_{- 3}^{2} x^{2} d x = \frac{35}{3}

\int_{- 3}^{2} 6 x d x = - 15

= - \frac{65}{4} + \frac{35}{3} - (- 15)

簡素化

- \frac{65}{4} + \frac{35}{3} - (- 15) : \frac{125}{12}

= \frac{125}{12}