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Beliebt Rechnen >

integral of xe^{(2x)/3}

Lösung

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

Lösung

\frac{9}{4} (\frac{2}{3} e^{\frac{2 x}{3}} x - e^{\frac{2 x}{3}}) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= \frac{9}{4} \cdot \int e^{u} u d u

Wende die partielle Integration an

= \frac{9}{4} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{9}{4} (e^{u} u - e^{u})

Setze in

u = \frac{2 x}{3}

ein

= \frac{9}{4} (e^{\frac{2 x}{3}} \frac{2 x}{3} - e^{\frac{2 x}{3}})

Vereinfache

\frac{9}{4} (e^{\frac{2 x}{3}} \frac{2 x}{3} - e^{\frac{2 x}{3}}) : \frac{9}{4} (\frac{2}{3} e^{\frac{2 x}{3}} x - e^{\frac{2 x}{3}})

= \frac{9}{4} (\frac{2}{3} e^{\frac{2 x}{3}} x - e^{\frac{2 x}{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{9}{4} (\frac{2}{3} e^{\frac{2 x}{3}} x - e^{\frac{2 x}{3}}) + C

Graph

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