integral of x(x+7)^7

Lösung

\int x (x + 7)^{7} d x

\frac{1}{72} (x + 7)^{8} (8 x - 7) + C

Schritte zur Lösung

\int x (x + 7)^{7} d x

Wende U-Substitution an

Entferne die Konstante:

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

Multipliziere aus

Wende die Summenregel an:

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

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

= \frac{1}{7} (\frac{7}{9} u^{\frac{9}{7}} - \frac{49}{8} u^{\frac{8}{7}})

Setze in

u = (x + 7)^{7}

ein

= \frac{1}{7} (\frac{7}{9} ((x + 7)^{7})^{\frac{9}{7}} - \frac{49}{8} ((x + 7)^{7})^{\frac{8}{7}})

Vereinfache

\frac{1}{7} (\frac{7}{9} ((x + 7)^{7})^{\frac{9}{7}} - \frac{49}{8} ((x + 7)^{7})^{\frac{8}{7}}) : \frac{1}{72} (x + 7)^{8} (8 x - 7)

= \frac{1}{72} (x + 7)^{8} (8 x - 7)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{72} (x + 7)^{8} (8 x - 7) + C