integral of x^5*e^{x^2}

Lösung

\int x^{5} \cdot e^{x^{2}} d x

\frac{1}{2} (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})) + C

Schritte zur Lösung

\int x^{5} e^{x^{2}} d x

Wende U-Substitution an

= \int \frac{e ^{u^{\frac{2}{5}}} u ^{\frac{1}{5}}}{5} d u

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int e^{u^{\frac{2}{5}}} u^{\frac{1}{5}} d u

Wende U-Substitution an

= \frac{1}{5} \cdot \int \frac{5 e ^{v} v ^{2}}{2} d v

Entferne die Konstante:

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

= \frac{1}{5} \cdot \frac{5}{2} \cdot \int e^{v} v^{2} d v

Wende die partielle Integration an

= \frac{1}{5} \cdot \frac{5}{2} (v^{2} e^{v} - \int 2 v e^{v} d v)

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

= \frac{1}{5} \cdot \frac{5}{2} (v^{2} e^{v} - 2 (e^{v} v - e^{v}))

Ersetze zurück

= \frac{1}{5} \cdot \frac{5}{2} (((x^{5})^{\frac{2}{5}})^{2} e^{(x^{5})^{\frac{2}{5}}} - 2 (e^{(x^{5})^{\frac{2}{5}}} (x^{5})^{\frac{2}{5}} - e^{(x^{5})^{\frac{2}{5}}}))

Vereinfache

\frac{1}{5} \cdot \frac{5}{2} (((x^{5})^{\frac{2}{5}})^{2} e^{(x^{5})^{\frac{2}{5}}} - 2 (e^{(x^{5})^{\frac{2}{5}}} (x^{5})^{\frac{2}{5}} - e^{(x^{5})^{\frac{2}{5}}})) : \frac{1}{2} (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}}))

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

Füge eine Konstante zur Lösung hinzu

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

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