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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

= 2 (\frac{3}{5} v^{\frac{5}{3}} - \frac{3}{2} v^{\frac{2}{3}})

Ersetze zurück

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

Vereinfache

2 (\frac{3}{5} (x + 1)^{\frac{5}{3}} - \frac{3}{2} (x + 1)^{\frac{2}{3}}) : \frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}}

= \frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}} + C

Graph

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