integral from 5 to 6 of x(x-5)^2

解

\int_{5}^{6} x (x - 5)^{2} d x

\frac{23}{12}

十進法表記

1.91666 \dots

解答ステップ

\int_{5}^{6} x (x - 5)^{2} d x

拡張

x (x - 5)^{2} : x^{3} - 10 x^{2} + 25 x

= \int_{5}^{6} x^{3} - 10 x^{2} + 25 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_{5}^{6} x^{3} d x - \int_{5}^{6} 10 x^{2} d x + \int_{5}^{6} 25 x d x

\int_{5}^{6} x^{3} d x = \frac{671}{4}

\int_{5}^{6} 10 x^{2} d x = \frac{910}{3}

\int_{5}^{6} 25 x d x = \frac{275}{2}

= \frac{671}{4} - \frac{910}{3} + \frac{275}{2}

簡素化

\frac{671}{4} - \frac{910}{3} + \frac{275}{2} : \frac{23}{12}

= \frac{23}{12}