integral from 1 to 3 of 10x(x-1)^3

解

\int_{1}^{3} 10 x (x - 1)^{3} d x

104

解答ステップ

\int_{1}^{3} 10 x (x - 1)^{3} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int_{1}^{3} x (x - 1)^{3} d x

拡張

x (x - 1)^{3} : x^{4} - 3 x^{3} + 3 x^{2} - x

= 10 \cdot \int_{1}^{3} x^{4} - 3 x^{3} + 3 x^{2} - 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

= 10 (\int_{1}^{3} x^{4} d x - \int_{1}^{3} 3 x^{3} d x + \int_{1}^{3} 3 x^{2} d x - \int_{1}^{3} x d x)

\int_{1}^{3} x^{4} d x = \frac{242}{5}

\int_{1}^{3} 3 x^{3} d x = 60

\int_{1}^{3} 3 x^{2} d x = 26

\int_{1}^{3} x d x = 4

= 10 (\frac{242}{5} - 60 + 26 - 4)

簡素化

10 (\frac{242}{5} - 60 + 26 - 4) : 104

= 104