integral from 2 to 10 of x(10-x)^2

解

\int_{2}^{10} x (10 - x)^{2} d x

\frac{2048}{3}

十進法表記

682.66666 \dots

解答ステップ

\int_{2}^{10} x (10 - x)^{2} d x

拡張

x (10 - x)^{2} : 100 x - 20 x^{2} + x^{3}

= \int_{2}^{10} 100 x - 20 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_{2}^{10} 100 x d x - \int_{2}^{10} 20 x^{2} d x + \int_{2}^{10} x^{3} d x

\int_{2}^{10} 100 x d x = 4800

\int_{2}^{10} 20 x^{2} d x = \frac{19840}{3}

\int_{2}^{10} x^{3} d x = 2496

= 4800 - \frac{19840}{3} + 2496

簡素化

4800 - \frac{19840}{3} + 2496 : \frac{2048}{3}

= \frac{2048}{3}