integral of sqrt(xe^{\sqrt{x)}}

Lösung

\int x e^{x} d x

2 (2 e^{\frac{1}{2} x} x - 16 (\frac{1}{2} e^{\frac{1}{2} x} x - e^{\frac{1}{2} x})) + C

Schritte zur Lösung

\int x e^{x} d x

Wende U-Substitution an

= \int 2 e^{\frac{u}{2}} u^{2} d u

Entferne die Konstante:

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

= 2 \cdot \int e^{\frac{u}{2}} u^{2} d u

Wende die partielle Integration an

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

\int 4 e^{\frac{u}{2}} u d u = 16 (\frac{1}{2} e^{\frac{u}{2}} u - e^{\frac{u}{2}})

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

Setze in

u = x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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