integral of (x^3)/3 e^{x^2}

Lösung

\int \frac{x ^{3}}{3} e^{x^{2}} d x

\frac{1}{6} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

Schritte zur Lösung

\int \frac{x ^{3}}{3} e^{x^{2}} d x

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int x^{3} e^{x^{2}} d x

Wende U-Substitution an

= \frac{1}{3} \cdot \int \frac{e ^{u^{\frac{2}{3}}} u ^{\frac{1}{3}}}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \frac{1}{3} \cdot \int e^{u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Wende U-Substitution an

= \frac{1}{3} \cdot \frac{1}{3} \cdot \int \frac{3 e ^{v} v}{2} d v

Entferne die Konstante:

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

= \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \int e^{v} v d v

Wende die partielle Integration an

= \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{3}{2} (e^{v} v - \int e^{v} d v)

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

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

Ersetze zurück

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

Vereinfache

\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{3}{2} (e^{(x^{3})^{\frac{2}{3}}} (x^{3})^{\frac{2}{3}} - e^{(x^{3})^{\frac{2}{3}}}) : \frac{1}{6} (e^{x^{2}} x^{2} - e^{x^{2}})

= \frac{1}{6} (e^{x^{2}} x^{2} - e^{x^{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

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