integral of x^{1/3}e^{x^{1/3}}

Lösung

\int x^{\frac{1}{3}} e^{x^{\frac{1}{3}}} d x

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

Schritte zur Lösung

\int x^{\frac{1}{3}} e^{x^{\frac{1}{3}}} d x

Wende U-Substitution an

= \int 3 e^{u} u^{3} d u

Entferne die Konstante:

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

= 3 \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= 3 (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= 3 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = x^{\frac{1}{3}}

ein

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

Vereinfache

3 ((x^{\frac{1}{3}})^{3} e^{x^{\frac{1}{3}}} - 3 ((x^{\frac{1}{3}})^{2} e^{x^{\frac{1}{3}}} - 2 (e^{x^{\frac{1}{3}}} x^{\frac{1}{3}} - e^{x^{\frac{1}{3}}}))) : 3 (e^{x^{\frac{1}{3}}} x - 3 (e^{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 2 (e^{x^{\frac{1}{3}}} x^{\frac{1}{3}} - e^{x^{\frac{1}{3}}})))

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

Füge eine Konstante zur Lösung hinzu

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

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