integral of 1/2 xe^{x/4}

Lösung

\int \frac{1}{2} x e^{\frac{x}{4}} d x

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

Schritte zur Lösung

\int \frac{1}{2} x e^{\frac{x}{4}} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int x e^{\frac{x}{4}} d x

Wende die partielle Integration an

= \frac{1}{2} (4 e^{\frac{x}{4}} x - \int 4 e^{\frac{x}{4}} d x)

\int 4 e^{\frac{x}{4}} d x = 16 e^{\frac{x}{4}}

= \frac{1}{2} (4 e^{\frac{x}{4}} x - 16 e^{\frac{x}{4}})

Vereinfache

\frac{1}{2} (4 e^{\frac{x}{4}} x - 16 e^{\frac{x}{4}}) : 2 e^{\frac{x}{4}} (x - 4)

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

Füge eine Konstante zur Lösung hinzu

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

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