integral of (x^5sqrt(1-x^2))

Lösung

\int (x^{5} 1 - x^{2}) d x

- \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + \frac{2}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{1}{7} (1 - x^{2})^{\frac{7}{2}} + C

Schritte zur Lösung

\int x^{5} 1 - x^{2} d x

Trigonometrische Substitution anwenden

= \int sin^{5} (u) cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (u))^{2} sin (u) cos^{2} (u) d u

Wende U-Substitution an

= \int - v^{2} (1 - v^{2})^{2} d v

Multipliziere aus

- v^{2} (1 - v^{2})^{2} : - v^{2} + 2 v^{4} - v^{6}

= \int - v^{2} + 2 v^{4} - v^{6} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int v^{2} d v + \int 2 v^{4} d v - \int v^{6} d v

\int v^{2} d v = \frac{v ^{3}}{3}

\int 2 v^{4} d v = \frac{2 v ^{5}}{5}

\int v^{6} d v = \frac{v ^{7}}{7}

= - \frac{v ^{3}}{3} + \frac{2 v ^{5}}{5} - \frac{v ^{7}}{7}

Ersetze zurück

= - \frac{cos ^{3} ( arcsin ( x ) )}{3} + \frac{2 cos ^{5} ( arcsin ( x ) )}{5} - \frac{cos ^{7} ( arcsin ( x ) )}{7}

Vereinfache

- \frac{cos ^{3} ( arcsin ( x ) )}{3} + \frac{2 cos ^{5} ( arcsin ( x ) )}{5} - \frac{cos ^{7} ( arcsin ( x ) )}{7} : - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + \frac{2}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{1}{7} (1 - x^{2})^{\frac{7}{2}}

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + \frac{2}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{1}{7} (1 - x^{2})^{\frac{7}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + \frac{2}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{1}{7} (1 - x^{2})^{\frac{7}{2}} + C

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