integral of e^{4x^2}(-4x^3)

Lösung

\int e^{4 x^{2}} (- 4 x^{3}) d x

- \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}}) + C

Schritte zur Lösung

\int e^{4 x^{2}} (- 4 x^{3}) d x

Vereinfache

= \int - 4 e^{4 x^{2}} x^{3} d x

Entferne die Konstante:

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

= - 4 \cdot \int e^{4 x^{2}} x^{3} d x

Wende U-Substitution an

= - 4 \cdot \int \frac{e ^{u} u}{32} d u

Entferne die Konstante:

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

= - 4 \cdot \frac{1}{32} \cdot \int e^{u} u d u

Wende die partielle Integration an

= - 4 \cdot \frac{1}{32} (e^{u} u - \int e^{u} d u)

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

= - 4 \cdot \frac{1}{32} (e^{u} u - e^{u})

Setze in

u = 4 x^{2}

ein

= - 4 \cdot \frac{1}{32} (e^{4 x^{2}} \cdot 4 x^{2} - e^{4 x^{2}})

Vereinfache

- 4 \cdot \frac{1}{32} (e^{4 x^{2}} \cdot 4 x^{2} - e^{4 x^{2}}) : - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}})

= - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}}) + C

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