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Beliebt Rechnen >

integral of (x^3)/(\sqrt[5]{x^2+9)}

Lösung

\int \frac{x ^{3}}{5 x ^{2} + 9} d x

Lösung

\frac{1}{2} (\frac{5}{9} (x^{2} + 9)^{\frac{9}{5}} - \frac{45}{4} (x^{2} + 9)^{\frac{4}{5}}) + C

Schritte zur Lösung

\int \frac{x ^{3}}{5 x ^{2} + 9} d x

Wende U-Substitution an

= \int \frac{u - 9}{2 5 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 - 9}{5 u} d u

Multipliziere aus

\frac{u - 9}{5 u} : u^{\frac{4}{5}} - \frac{9}{5 u}

= \frac{1}{2} \cdot \int u^{\frac{4}{5}} - \frac{9}{5 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{4}{5}} d u - \int \frac{9}{5 u} d u)

\int u^{\frac{4}{5}} d u = \frac{5}{9} u^{\frac{9}{5}}

\int \frac{9}{5 u} d u = \frac{45}{4} u^{\frac{4}{5}}

= \frac{1}{2} (\frac{5}{9} u^{\frac{9}{5}} - \frac{45}{4} u^{\frac{4}{5}})

Setze in

u = x^{2} + 9

ein

= \frac{1}{2} (\frac{5}{9} (x^{2} + 9)^{\frac{9}{5}} - \frac{45}{4} (x^{2} + 9)^{\frac{4}{5}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{5}{9} (x^{2} + 9)^{\frac{9}{5}} - \frac{45}{4} (x^{2} + 9)^{\frac{4}{5}}) + C

Graph

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