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Beliebt Rechnen >

integral of x/((2x-1)^{2/3)}

Lösung

\int \frac{x}{( 2 x - 1 ) ^{\frac{2}{3}}} d x

Lösung

\frac{1}{4} (\frac{3}{4} (2 x - 1)^{\frac{4}{3}} + 3 (2 x - 1)^{\frac{1}{3}}) + C

Schritte zur Lösung

\int \frac{x}{( 2 x - 1 ) ^{\frac{2}{3}}} d x

Wende U-Substitution an

= \int \frac{u + 1}{4 u ^{\frac{2}{3}}} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int \frac{u + 1}{u ^{\frac{2}{3}}} d u

Multipliziere aus

\frac{u + 1}{u ^{\frac{2}{3}}} : u^{\frac{1}{3}} + \frac{1}{u ^{\frac{2}{3}}}

= \frac{1}{4} \cdot \int u^{\frac{1}{3}} + \frac{1}{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

= \frac{1}{4} (\int u^{\frac{1}{3}} d u + \int \frac{1}{u ^{\frac{2}{3}}} d u)

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

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

= \frac{1}{4} (\frac{3}{4} u^{\frac{4}{3}} + 3 u^{\frac{1}{3}})

Setze in

u = 2 x - 1

ein

= \frac{1}{4} (\frac{3}{4} (2 x - 1)^{\frac{4}{3}} + 3 (2 x - 1)^{\frac{1}{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (\frac{3}{4} (2 x - 1)^{\frac{4}{3}} + 3 (2 x - 1)^{\frac{1}{3}}) + C

Graph

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