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

Lösung

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

\frac{1}{7} \cdot \frac{7}{2} (((x^{7})^{\frac{2}{7}})^{3} e^{(x^{7})^{\frac{2}{7}}} - 3 (((x^{7})^{\frac{2}{7}})^{2} e^{(x^{7})^{\frac{2}{7}}} - 2 (e^{(x^{7})^{\frac{2}{7}}} (x^{7})^{\frac{2}{7}} - e^{(x^{7})^{\frac{2}{7}}}))) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= \frac{1}{7} \cdot \int \frac{7 e ^{v} v ^{3}}{2} d v

Entferne die Konstante:

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

= \frac{1}{7} \cdot \frac{7}{2} \cdot \int e^{v} v^{3} d v

Wende die partielle Integration an

= \frac{1}{7} \cdot \frac{7}{2} (v^{3} e^{v} - \int 3 v^{2} e^{v} d v)

\int 3 v^{2} e^{v} d v = 3 (v^{2} e^{v} - 2 (e^{v} v - e^{v}))

= \frac{1}{7} \cdot \frac{7}{2} (v^{3} e^{v} - 3 (v^{2} e^{v} - 2 (e^{v} v - e^{v})))

Ersetze zurück

= \frac{1}{7} \cdot \frac{7}{2} (((x^{7})^{\frac{2}{7}})^{3} e^{(x^{7})^{\frac{2}{7}}} - 3 (((x^{7})^{\frac{2}{7}})^{2} e^{(x^{7})^{\frac{2}{7}}} - 2 (e^{(x^{7})^{\frac{2}{7}}} (x^{7})^{\frac{2}{7}} - e^{(x^{7})^{\frac{2}{7}}})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} \cdot \frac{7}{2} (((x^{7})^{\frac{2}{7}})^{3} e^{(x^{7})^{\frac{2}{7}}} - 3 (((x^{7})^{\frac{2}{7}})^{2} e^{(x^{7})^{\frac{2}{7}}} - 2 (e^{(x^{7})^{\frac{2}{7}}} (x^{7})^{\frac{2}{7}} - e^{(x^{7})^{\frac{2}{7}}}))) + C

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