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

Lösung

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

x^{\frac{2}{3}} (x + 1)^{3} - \frac{1}{910} (1092 x^{\frac{5}{3}} + 490 x^{\frac{13}{6}} + 390 x^{\frac{7}{6}} - 455 x^{\frac{2}{3}}) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= x^{\frac{2}{3}} (x + 1)^{3} - \int \frac{9 x ( x + 1 ) ^{2} - 2 ( x + 1 ) ^{3}}{6 x ^{\frac{1}{3}}} d x

\int \frac{9 x ( x + 1 ) ^{2} - 2 ( x + 1 ) ^{3}}{6 x ^{\frac{1}{3}}} d x = \frac{1}{910} (1092 x^{\frac{5}{3}} + 490 x^{\frac{13}{6}} + 390 x^{\frac{7}{6}} - 455 x^{\frac{2}{3}})

= x^{\frac{2}{3}} (x + 1)^{3} - \frac{1}{910} (1092 x^{\frac{5}{3}} + 490 x^{\frac{13}{6}} + 390 x^{\frac{7}{6}} - 455 x^{\frac{2}{3}})

Füge eine Konstante zur Lösung hinzu

= x^{\frac{2}{3}} (x + 1)^{3} - \frac{1}{910} (1092 x^{\frac{5}{3}} + 490 x^{\frac{13}{6}} + 390 x^{\frac{7}{6}} - 455 x^{\frac{2}{3}}) + C

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