integral from x to 5 of (6(y-x))/(125)

解

\int_{x}^{5} \frac{6 ( y - x )}{125} d y

\frac{3 ( x ^{2} - 10 x + 25 )}{125}

解答ステップ

\int_{x}^{5} \frac{6 ( y - x )}{125} d y

定数を除く:

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

= \frac{6}{125} \cdot \int_{x}^{5} - x + y d y

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_{x}^{5} x d y + \int_{x}^{5} y d y)

\int_{x}^{5} x d y = x \cdot 5 - x^{2}

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

= \frac{6}{125} (- (x \cdot 5 - x^{2}) + \frac{25 - x ^{2}}{2})

簡素化

\frac{6}{125} (- (x \cdot 5 - x^{2}) + \frac{25 - x ^{2}}{2}) : \frac{3 ( x ^{2} - 10 x + 25 )}{125}

= \frac{3 ( x ^{2} - 10 x + 25 )}{125}