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Beliebt Rechnen >

integral of x^2\sqrt[3]{x-1}

Lösung

\int x^{2} 3 x - 1 d x

Lösung

\frac{3}{10} (x - 1)^{\frac{10}{3}} + \frac{6}{7} (x - 1)^{\frac{7}{3}} + \frac{3}{4} (x - 1)^{\frac{4}{3}} + C

Schritte zur Lösung

\int x^{2} 3 x - 1 d x

Wende U-Substitution an

= \int (u + 1)^{2} 3 u d u

Multipliziere aus

(u + 1)^{2} 3 u : u^{\frac{7}{3}} + 2 u^{\frac{4}{3}} + 3 u

= \int u^{\frac{7}{3}} + 2 u^{\frac{4}{3}} + 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{7}{3}} d u + \int 2 u^{\frac{4}{3}} d u + \int 3 u d u

\int u^{\frac{7}{3}} d u = \frac{3}{10} u^{\frac{10}{3}}

\int 2 u^{\frac{4}{3}} d u = \frac{6}{7} u^{\frac{7}{3}}

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

= \frac{3}{10} u^{\frac{10}{3}} + \frac{6}{7} u^{\frac{7}{3}} + \frac{3}{4} u^{\frac{4}{3}}

Setze in

u = x - 1

ein

= \frac{3}{10} (x - 1)^{\frac{10}{3}} + \frac{6}{7} (x - 1)^{\frac{7}{3}} + \frac{3}{4} (x - 1)^{\frac{4}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{10} (x - 1)^{\frac{10}{3}} + \frac{6}{7} (x - 1)^{\frac{7}{3}} + \frac{3}{4} (x - 1)^{\frac{4}{3}} + C

Graph

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