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

Lösung

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

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

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= \int - 1 + 2 v^{2} - v^{4} 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 1 d v + \int 2 v^{2} d v - \int v^{4} d v

\int 1 d v = v

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

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

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

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e 4^{x + 1}

x \to 1 lim (2 x^{3} - 4 x^{2} + x + 7)

\frac{\partial}{\partial x} (cos (2 y))

\int 39 - 10 x - x^{2} d x

\frac{d}{d x} (18 x - x^{2})