integral from 10 to x of (-1)/(75)(x-20)

解

\int_{10}^{x} \frac{- 1}{75} (x - 20) d x

- \frac{1}{150} (x^{2} - 40 x + 300)

解答ステップ

\int_{10}^{x} \frac{- 1}{75} (x - 20) d x

定数を除く:

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

= \frac{- 1}{75} \cdot \int_{10}^{x} x - 20 d x

簡素化

= (- \frac{1}{75}) \cdot \int_{10}^{x} x - 20 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{1}{75}) (\int_{10}^{x} x d x - \int_{10}^{x} 20 d x)

\int_{10}^{x} x d x = \frac{x ^{2}}{2} - 50

\int_{10}^{x} 20 d x = 20 x - 200

= (- \frac{1}{75}) (\frac{x ^{2}}{2} - 50 - (20 x - 200))

簡素化

(- \frac{1}{75}) (\frac{x ^{2}}{2} - 50 - (20 x - 200)) : - \frac{1}{150} (x^{2} - 40 x + 300)

= - \frac{1}{150} (x^{2} - 40 x + 300)