integral of x*(7-x)^9

Lösung

\int x \cdot (7 - x)^{9} d x

- \frac{1}{110} (- x + 7)^{10} (10 x + 7) + C

Schritte zur Lösung

\int x (7 - x)^{9} d x

Wende U-Substitution an

= \int - \frac{u ^{\frac{1}{9}} ( - 9 u + 7 )}{9} d u

Entferne die Konstante:

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

= - \frac{1}{9} \cdot \int u^{\frac{1}{9}} (- 9 u + 7) d u

Multipliziere aus

u^{\frac{1}{9}} (- 9 u + 7) : - u^{\frac{2}{9}} + 7 u^{\frac{1}{9}}

= - \frac{1}{9} \cdot \int - u^{\frac{2}{9}} + 7 u^{\frac{1}{9}} 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}{9} (- \int u^{\frac{2}{9}} d u + \int 7 u^{\frac{1}{9}} d u)

\int u^{\frac{2}{9}} d u = \frac{9}{11} u^{\frac{11}{9}}

\int 7 u^{\frac{1}{9}} d u = \frac{63}{10} u^{\frac{10}{9}}

= - \frac{1}{9} (- \frac{9}{11} u^{\frac{11}{9}} + \frac{63}{10} u^{\frac{10}{9}})

Setze in

u = (7 - x)^{9}

ein

= - \frac{1}{9} (- \frac{9}{11} ((7 - x)^{9})^{\frac{11}{9}} + \frac{63}{10} ((7 - x)^{9})^{\frac{10}{9}})

Vereinfache

- \frac{1}{9} (- \frac{9}{11} ((7 - x)^{9})^{\frac{11}{9}} + \frac{63}{10} ((7 - x)^{9})^{\frac{10}{9}}) : - \frac{1}{110} (- x + 7)^{10} (10 x + 7)

= - \frac{1}{110} (- x + 7)^{10} (10 x + 7)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{110} (- x + 7)^{10} (10 x + 7) + C

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