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Beliebt Rechnen >

integral of 4xe^{(4x)/3}

Lösung

\int 4 x e^{\frac{4 x}{3}} d x

Lösung

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

Schritte zur Lösung

\int 4 x e^{\frac{4 x}{3}} d x

Entferne die Konstante:

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

= 4 \cdot \int x e^{\frac{4 x}{3}} d x

Wende U-Substitution an

= 4 \cdot \int \frac{9 e ^{u} u}{16} d u

Entferne die Konstante:

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

= 4 \cdot \frac{9}{16} \cdot \int e^{u} u d u

Wende die partielle Integration an

= 4 \cdot \frac{9}{16} (e^{u} u - \int e^{u} d u)

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

= 4 \cdot \frac{9}{16} (e^{u} u - e^{u})

Setze in

u = \frac{4 x}{3}

ein

= 4 \cdot \frac{9}{16} (e^{\frac{4 x}{3}} \frac{4 x}{3} - e^{\frac{4 x}{3}})

Vereinfache

4 \cdot \frac{9}{16} (e^{\frac{4 x}{3}} \frac{4 x}{3} - e^{\frac{4 x}{3}}) : 3 e^{\frac{4 x}{3}} x - \frac{9}{4} e^{\frac{4 x}{3}}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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