integral of (x^6)/(sqrt(x^{14)-7)}

Lösung

\int \frac{x ^{6}}{x ^{14} - 7} d x

\frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7} + C

Schritte zur Lösung

\int \frac{x ^{6}}{x ^{14} - 7} d x

Wende U-Substitution an

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

Trigonometrische Substitution anwenden

= \frac{1}{7} \cdot \int sec (v) d v

Nutze das gemeinsame Integral :

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{1}{7} ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

= \frac{1}{7} ln tan (arcsec (\frac{1}{7} x^{7})) + sec (arcsec (\frac{1}{7} x^{7}))

Vereinfache

\frac{1}{7} ln tan (arcsec (\frac{1}{7} x^{7})) + sec (arcsec (\frac{1}{7} x^{7})) : \frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7}

= \frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7} + C

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