integral of x^3e^{2x^2}

Lösung

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

\frac{1}{8} (e^{2 x^{2}} \cdot 2 x^{2} - e^{2 x^{2}}) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{e ^{u} u}{8} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int e^{u} u d u

Wende die partielle Integration an

= \frac{1}{8} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{8} (e^{u} u - e^{u})

Setze in

u = 2 x^{2}

ein

= \frac{1}{8} (e^{2 x^{2}} \cdot 2 x^{2} - e^{2 x^{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (e^{2 x^{2}} \cdot 2 x^{2} - e^{2 x^{2}}) + C

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