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Beliebt Rechnen >

integral of x/((x+6)^{2/3)}

Lösung

\int \frac{x}{( x + 6 ) ^{\frac{2}{3}}} d x

Lösung

\frac{3}{4} (x + 6)^{\frac{4}{3}} - 18 (x + 6)^{\frac{1}{3}} + C

Schritte zur Lösung

\int \frac{x}{( x + 6 ) ^{\frac{2}{3}}} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{u - 6}{u ^{\frac{2}{3}}} : u^{\frac{1}{3}} - \frac{6}{u ^{\frac{2}{3}}}

= \int u^{\frac{1}{3}} - \frac{6}{u ^{\frac{2}{3}}} 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{1}{3}} d u - \int \frac{6}{u ^{\frac{2}{3}}} d u

\int u^{\frac{1}{3}} d u = \frac{3}{4} u^{\frac{4}{3}}

\int \frac{6}{u ^{\frac{2}{3}}} d u = 18 u^{\frac{1}{3}}

= \frac{3}{4} u^{\frac{4}{3}} - 18 u^{\frac{1}{3}}

Setze in

u = x + 6

ein

= \frac{3}{4} (x + 6)^{\frac{4}{3}} - 18 (x + 6)^{\frac{1}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{4} (x + 6)^{\frac{4}{3}} - 18 (x + 6)^{\frac{1}{3}} + C

Graph

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