integral from 5 to x of 2/75 (x-5)

解

\int_{5}^{x} \frac{2}{75} (x - 5) d x

\frac{1}{75} (x^{2} - 10 x + 25)

解答ステップ

\int_{5}^{x} \frac{2}{75} (x - 5) d x

定数を除く:

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

= \frac{2}{75} \cdot \int_{5}^{x} x - 5 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

= \frac{2}{75} (\int_{5}^{x} x d x - \int_{5}^{x} 5 d x)

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

\int_{5}^{x} 5 d x = 5 x - 25

= \frac{2}{75} (\frac{x ^{2} - 25}{2} - (5 x - 25))

簡素化

\frac{2}{75} (\frac{x ^{2} - 25}{2} - (5 x - 25)) : \frac{1}{75} (x^{2} - 10 x + 25)

= \frac{1}{75} (x^{2} - 10 x + 25)