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Beliebt Rechnen >

integral of ((1+\sqrt[3]{x})^2)/(\sqrt[3]{x)}

Lösung

\int \frac{( 1 + 3 x ) ^{2}}{3 x} d x

Lösung

\frac{3}{4} x^{\frac{4}{3}} + \frac{3}{2} x^{\frac{2}{3}} + 2 x^{\frac{2}{3}} 3 x + C

Schritte zur Lösung

\int \frac{( 1 + 3 x ) ^{2}}{3 x} d x

Wende die partielle Integration an

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

\int \frac{x ^{\frac{2}{3}} - 1}{3 x ^{\frac{1}{3}}} d x = \frac{1}{4} (x^{\frac{4}{3}} - 2 x^{\frac{2}{3}})

= x^{\frac{2}{3}} (3 x + 1)^{2} - \frac{1}{4} (x^{\frac{4}{3}} - 2 x^{\frac{2}{3}})

Vereinfache

x^{\frac{2}{3}} (3 x + 1)^{2} - \frac{1}{4} (x^{\frac{4}{3}} - 2 x^{\frac{2}{3}}) : \frac{3}{4} x^{\frac{4}{3}} + \frac{3}{2} x^{\frac{2}{3}} + 2 x^{\frac{2}{3}} 3 x

= \frac{3}{4} x^{\frac{4}{3}} + \frac{3}{2} x^{\frac{2}{3}} + 2 x^{\frac{2}{3}} 3 x

Füge eine Konstante zur Lösung hinzu

= \frac{3}{4} x^{\frac{4}{3}} + \frac{3}{2} x^{\frac{2}{3}} + 2 x^{\frac{2}{3}} 3 x + C

Graph

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