integral of 2x^{13}e^{x^7}

Lösung

\int 2 x^{13} e^{x^{7}} d x

\frac{2}{7} (e^{x^{7}} x^{7} - e^{x^{7}}) + C

Schritte zur Lösung

\int 2 x^{13} e^{x^{7}} d x

Entferne die Konstante:

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

= 2 \cdot \int x^{13} e^{x^{7}} d x

Wende U-Substitution an

= 2 \cdot \int \frac{e ^{u^{\frac{7}{13}}} u ^{\frac{1}{13}}}{13} d u

Entferne die Konstante:

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

= 2 \cdot \frac{1}{13} \cdot \int e^{u^{\frac{7}{13}}} u^{\frac{1}{13}} d u

Wende U-Substitution an

= 2 \cdot \frac{1}{13} \cdot \int \frac{13 e ^{v} v}{7} d v

Entferne die Konstante:

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

= 2 \cdot \frac{1}{13} \cdot \frac{13}{7} \cdot \int e^{v} v d v

Wende die partielle Integration an

= 2 \cdot \frac{1}{13} \cdot \frac{13}{7} (e^{v} v - \int e^{v} d v)

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

= 2 \cdot \frac{1}{13} \cdot \frac{13}{7} (e^{v} v - e^{v})

Ersetze zurück

= 2 \cdot \frac{1}{13} \cdot \frac{13}{7} (e^{(x^{13})^{\frac{7}{13}}} (x^{13})^{\frac{7}{13}} - e^{(x^{13})^{\frac{7}{13}}})

Vereinfache

2 \cdot \frac{1}{13} \cdot \frac{13}{7} (e^{(x^{13})^{\frac{7}{13}}} (x^{13})^{\frac{7}{13}} - e^{(x^{13})^{\frac{7}{13}}}) : \frac{2}{7} (e^{x^{7}} x^{7} - e^{x^{7}})

= \frac{2}{7} (e^{x^{7}} x^{7} - e^{x^{7}})

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} (e^{x^{7}} x^{7} - e^{x^{7}}) + C

x \to 0 lim (\frac{2 ^{x} - 1}{2})

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