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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{3}{8} (x + 3)^{\frac{8}{3}} - \frac{18}{5} (x + 3)^{\frac{5}{3}} + \frac{27}{2} (x + 3)^{\frac{2}{3}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( u - 3 ) ^{2}}{3 u} : u^{\frac{5}{3}} - 6 u^{\frac{2}{3}} + \frac{9}{3 u}

= \int u^{\frac{5}{3}} - 6 u^{\frac{2}{3}} + \frac{9}{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

= \int u^{\frac{5}{3}} d u - \int 6 u^{\frac{2}{3}} d u + \int \frac{9}{3 u} d u

\int u^{\frac{5}{3}} d u = \frac{3}{8} u^{\frac{8}{3}}

\int 6 u^{\frac{2}{3}} d u = \frac{18}{5} u^{\frac{5}{3}}

\int \frac{9}{3 u} d u = \frac{27}{2} u^{\frac{2}{3}}

= \frac{3}{8} u^{\frac{8}{3}} - \frac{18}{5} u^{\frac{5}{3}} + \frac{27}{2} u^{\frac{2}{3}}

Setze in

u = x + 3

ein

= \frac{3}{8} (x + 3)^{\frac{8}{3}} - \frac{18}{5} (x + 3)^{\frac{5}{3}} + \frac{27}{2} (x + 3)^{\frac{2}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{8} (x + 3)^{\frac{8}{3}} - \frac{18}{5} (x + 3)^{\frac{5}{3}} + \frac{27}{2} (x + 3)^{\frac{2}{3}} + C

Graph

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