integral from 0 to y of (6(y-x))/(125)

解

\int_{0}^{y} \frac{6 ( y - x )}{125} d x

\frac{3 y ^{2}}{125}

解答ステップ

\int_{0}^{y} \frac{6 ( y - x )}{125} d x

定数を除く:

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

= \frac{6}{125} \cdot \int_{0}^{y} y - 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

= \frac{6}{125} (\int_{0}^{y} y d x - \int_{0}^{y} x d x)

\int_{0}^{y} y d x = y^{2}

\int_{0}^{y} x d x = \frac{y ^{2}}{2}

= \frac{6}{125} (y^{2} - \frac{y ^{2}}{2})

簡素化

\frac{6}{125} (y^{2} - \frac{y ^{2}}{2}) : \frac{3 y ^{2}}{125}

= \frac{3 y ^{2}}{125}