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

Lösung

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

\frac{1}{5} (x + 2) (3 x + 1)^{\frac{2}{3}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

= \int \frac{v + 2}{9 3 v} d v

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int \frac{v + 2}{3 v} d v

Multipliziere aus

\frac{v + 2}{3 v} : v^{\frac{2}{3}} + \frac{2}{3 v}

= \frac{1}{9} \cdot \int v^{\frac{2}{3}} + \frac{2}{3 v} d v

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 v^{\frac{2}{3}} d v + \int \frac{2}{3 v} d v)

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

\int \frac{2}{3 v} d v = 3 v^{\frac{2}{3}}

= \frac{1}{9} (\frac{3}{5} v^{\frac{5}{3}} + 3 v^{\frac{2}{3}})

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (x + 2) (3 x + 1)^{\frac{2}{3}} + C

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