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Beliebt Rechnen >

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

Lösung

\int \frac{z ^{3}}{3 z ^{2} + 1} d z

Lösung

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

Schritte zur Lösung

\int \frac{z ^{3}}{3 z ^{2} + 1} d z

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Setze in

u = z^{2} + 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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