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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende Exponentenregel an:

\frac{1}{a ^{b}} = a^{- b}

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Setze in

u = - x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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