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

Lösung

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

e^{\frac{x ^{2}}{2}} x^{2} - 2 e^{\frac{x ^{2}}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Wende U-Substitution an

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int 4 e^{w} w d w

Entferne die Konstante:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 \cdot \int e^{w} w d w

Wende die partielle Integration an

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 (e^{w} w - \int e^{w} d w)

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 (e^{w} w - e^{w})

Ersetze zurück

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 e^{\frac{( x ^{3} ) ^{\frac{2}{3}}}{2}} \frac{( x ^{3} ) ^{\frac{2}{3}}}{2} - e^{\frac{( x ^{3} ) ^{\frac{2}{3}}}{2}}

Vereinfache

\frac{1}{3} \cdot \frac{3}{2} \cdot 4 e^{\frac{( x ^{3} ) ^{\frac{2}{3}}}{2}} \frac{( x ^{3} ) ^{\frac{2}{3}}}{2} - e^{\frac{( x ^{3} ) ^{\frac{2}{3}}}{2}} : e^{\frac{x ^{2}}{2}} x^{2} - 2 e^{\frac{x ^{2}}{2}}

= e^{\frac{x ^{2}}{2}} x^{2} - 2 e^{\frac{x ^{2}}{2}}

Füge eine Konstante zur Lösung hinzu

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

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