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Beliebt Rechnen >

integral of x/(\sqrt[3]{1-3x)}

Lösung

\int \frac{x}{3 1 - 3 x} d x

Lösung

- \frac{1}{10} (2 x + 1) (- 3 x + 1)^{\frac{2}{3}} + C

Schritte zur Lösung

\int \frac{x}{3 1 - 3 x} d x

Wende U-Substitution an

= \int \frac{u - 1}{9 3 u} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int \frac{u - 1}{3 u} d u

Multipliziere aus

\frac{u - 1}{3 u} : u^{\frac{2}{3}} - \frac{1}{3 u}

= \frac{1}{9} \cdot \int u^{\frac{2}{3}} - \frac{1}{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

= \frac{1}{9} (\int u^{\frac{2}{3}} d u - \int \frac{1}{3 u} d u)

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

\int \frac{1}{3 u} d u = \frac{3}{2} u^{\frac{2}{3}}

= \frac{1}{9} (\frac{3}{5} u^{\frac{5}{3}} - \frac{3}{2} u^{\frac{2}{3}})

Setze in

u = 1 - 3 x

ein

= \frac{1}{9} (\frac{3}{5} (1 - 3 x)^{\frac{5}{3}} - \frac{3}{2} (1 - 3 x)^{\frac{2}{3}})

Vereinfache

\frac{1}{9} (\frac{3}{5} (1 - 3 x)^{\frac{5}{3}} - \frac{3}{2} (1 - 3 x)^{\frac{2}{3}}) : - \frac{1}{10} (2 x + 1) (- 3 x + 1)^{\frac{2}{3}}

= - \frac{1}{10} (2 x + 1) (- 3 x + 1)^{\frac{2}{3}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{10} (2 x + 1) (- 3 x + 1)^{\frac{2}{3}} + C

Graph

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