integral from 0 to 3 of x(3x-x^2)

解

\int_{0}^{3} x (3 x - x^{2}) d x

\frac{27}{4}

十進法表記

6.75

解答ステップ

\int_{0}^{3} x (3 x - x^{2}) d x

拡張

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

= \int_{0}^{3} 3 x^{2} - 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

= \int_{0}^{3} 3 x^{2} d x - \int_{0}^{3} x^{3} d x

\int_{0}^{3} 3 x^{2} d x = 27

\int_{0}^{3} x^{3} d x = \frac{81}{4}

= 27 - \frac{81}{4}

簡素化

27 - \frac{81}{4} : \frac{27}{4}

= \frac{27}{4}