integral of x^5arccos(sqrt(1-x^6))

Lösung

\int x^{5} arccos (1 - x^{6}) d x

- \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6})))) + C

Schritte zur Lösung

\int x^{5} arccos (1 - x^{6}) d x

Wende U-Substitution an

= \int - \frac{arccos ( u )}{6} d u

Entferne die Konstante:

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

= - \frac{1}{6} \cdot \int arccos (u) d u

Wende U-Substitution an

= - \frac{1}{6} \cdot \int arccos (v) \cdot 2 v d v

Entferne die Konstante:

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

= - \frac{1}{6} \cdot 2 \cdot \int arccos (v) v d v

Wende die partielle Integration an

= - \frac{1}{6} \cdot 2 (\frac{1}{2} v^{2} arccos (v) - \int - \frac{v ^{2}}{2 1 - v ^{2}} d v)

\int - \frac{v ^{2}}{2 1 - v ^{2}} d v = - \frac{1}{4} (arcsin (v) - \frac{1}{2} sin (2 arcsin (v)))

= - \frac{1}{6} \cdot 2 (\frac{1}{2} v^{2} arccos (v) - (- \frac{1}{4} (arcsin (v) - \frac{1}{2} sin (2 arcsin (v)))))

Ersetze zurück

= - \frac{1}{6} \cdot 2 (\frac{1}{2} (1 - x^{6})^{2} arccos (1 - x^{6}) - (- \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6})))))

Vereinfache

- \frac{1}{6} \cdot 2 (\frac{1}{2} (1 - x^{6})^{2} arccos (1 - x^{6}) - (- \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6}))))) : - \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6}))))

= - \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6}))))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6})))) + C

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