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Beliebt Rechnen >

integral of x(x+3)^7

Lösung

\int x (x + 3)^{7} d x

Lösung

\frac{1}{72} (x + 3)^{8} (8 x - 3) + C

Schritte zur Lösung

\int x (x + 3)^{7} d x

Wende U-Substitution an

= \int \frac{7 u ( 7 u - 3 )}{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 7 u (7 u - 3) d u

Multipliziere aus

7 u (7 u - 3) : u^{\frac{2}{7}} - 3 7 u

= \frac{1}{7} \cdot \int u^{\frac{2}{7}} - 3 7 u 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 3 7 u d u)

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

\int 3 7 u d u = \frac{21}{8} u^{\frac{8}{7}}

= \frac{1}{7} (\frac{7}{9} u^{\frac{9}{7}} - \frac{21}{8} u^{\frac{8}{7}})

Setze in

u = (x + 3)^{7}

ein

= \frac{1}{7} (\frac{7}{9} ((x + 3)^{7})^{\frac{9}{7}} - \frac{21}{8} ((x + 3)^{7})^{\frac{8}{7}})

Vereinfache

\frac{1}{7} (\frac{7}{9} ((x + 3)^{7})^{\frac{9}{7}} - \frac{21}{8} ((x + 3)^{7})^{\frac{8}{7}}) : \frac{1}{72} (x + 3)^{8} (8 x - 3)

= \frac{1}{72} (x + 3)^{8} (8 x - 3)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{72} (x + 3)^{8} (8 x - 3) + C

Graph

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