integral of 14x^{15}e^{x^8}

Lösung

\int 14 x^{15} e^{x^{8}} d x

\frac{7}{4} (e^{x^{8}} x^{8} - e^{x^{8}}) + C

Schritte zur Lösung

\int 14 x^{15} e^{x^{8}} d x

Entferne die Konstante:

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

= 14 \cdot \int x^{15} e^{x^{8}} d x

Wende U-Substitution an

= 14 \cdot \int \frac{e ^{u^{\frac{8}{15}}} u ^{\frac{1}{15}}}{15} d u

Entferne die Konstante:

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

= 14 \cdot \frac{1}{15} \cdot \int e^{u^{\frac{8}{15}}} u^{\frac{1}{15}} d u

Wende U-Substitution an

= 14 \cdot \frac{1}{15} \cdot \int \frac{15 e ^{v} v}{8} d v

Entferne die Konstante:

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

= 14 \cdot \frac{1}{15} \cdot \frac{15}{8} \cdot \int e^{v} v d v

Wende die partielle Integration an

= 14 \cdot \frac{1}{15} \cdot \frac{15}{8} (e^{v} v - \int e^{v} d v)

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

= 14 \cdot \frac{1}{15} \cdot \frac{15}{8} (e^{v} v - e^{v})

Ersetze zurück

= 14 \cdot \frac{1}{15} \cdot \frac{15}{8} (e^{(x^{15})^{\frac{8}{15}}} (x^{15})^{\frac{8}{15}} - e^{(x^{15})^{\frac{8}{15}}})

Vereinfache

14 \cdot \frac{1}{15} \cdot \frac{15}{8} (e^{(x^{15})^{\frac{8}{15}}} (x^{15})^{\frac{8}{15}} - e^{(x^{15})^{\frac{8}{15}}}) : \frac{7}{4} (e^{x^{8}} x^{8} - e^{x^{8}})

= \frac{7}{4} (e^{x^{8}} x^{8} - e^{x^{8}})

Füge eine Konstante zur Lösung hinzu

= \frac{7}{4} (e^{x^{8}} x^{8} - e^{x^{8}}) + C

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