integral of x(5-x)^7

Lösung

\int x (5 - x)^{7} d x

- \frac{1}{72} (- x + 5)^{8} (8 x + 5) + C

Schritte zur Lösung

\int x (5 - x)^{7} d x

Wende U-Substitution an

= \int - \frac{u ^{\frac{1}{7}} ( - 7 u + 5 )}{7} d u

Entferne die Konstante:

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

= - \frac{1}{7} \cdot \int u^{\frac{1}{7}} (- 7 u + 5) d u

Multipliziere aus

u^{\frac{1}{7}} (- 7 u + 5) : - u^{\frac{2}{7}} + 5 u^{\frac{1}{7}}

= - \frac{1}{7} \cdot \int - u^{\frac{2}{7}} + 5 u^{\frac{1}{7}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{7} (- \int u^{\frac{2}{7}} d u + \int 5 u^{\frac{1}{7}} d u)

\int u^{\frac{2}{7}} d u = \frac{7}{9} u^{\frac{9}{7}}

\int 5 u^{\frac{1}{7}} d u = \frac{35}{8} u^{\frac{8}{7}}

= - \frac{1}{7} (- \frac{7}{9} u^{\frac{9}{7}} + \frac{35}{8} u^{\frac{8}{7}})

Setze in

u = (5 - x)^{7}

ein

= - \frac{1}{7} (- \frac{7}{9} ((5 - x)^{7})^{\frac{9}{7}} + \frac{35}{8} ((5 - x)^{7})^{\frac{8}{7}})

Vereinfache

- \frac{1}{7} (- \frac{7}{9} ((5 - x)^{7})^{\frac{9}{7}} + \frac{35}{8} ((5 - x)^{7})^{\frac{8}{7}}) : - \frac{1}{72} (- x + 5)^{8} (8 x + 5)

= - \frac{1}{72} (- x + 5)^{8} (8 x + 5)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{72} (- x + 5)^{8} (8 x + 5) + C

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