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Beliebt Rechnen >

integral of x/(\sqrt[3]{x+8)}

Lösung

\int \frac{x}{3 x + 8} d x

Lösung

\frac{3}{5} (x + 8)^{\frac{5}{3}} - 12 (x + 8)^{\frac{2}{3}} + C

Schritte zur Lösung

\int \frac{x}{3 x + 8} d x

Wende U-Substitution an

= \int \frac{u - 8}{3 u} d u

Multipliziere aus

\frac{u - 8}{3 u} : u^{\frac{2}{3}} - \frac{8}{3 u}

= \int u^{\frac{2}{3}} - \frac{8}{3 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

= \int u^{\frac{2}{3}} d u - \int \frac{8}{3 u} d u

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

\int \frac{8}{3 u} d u = 12 u^{\frac{2}{3}}

= \frac{3}{5} u^{\frac{5}{3}} - 12 u^{\frac{2}{3}}

Setze in

u = x + 8

ein

= \frac{3}{5} (x + 8)^{\frac{5}{3}} - 12 (x + 8)^{\frac{2}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{5} (x + 8)^{\frac{5}{3}} - 12 (x + 8)^{\frac{2}{3}} + C

Graph

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