integral of (1-x^2)^{3/2}x^3

Lösung

\int (1 - x^{2})^{\frac{3}{2}} x^{3} d x

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

Schritte zur Lösung

\int (1 - x^{2})^{\frac{3}{2}} x^{3} d x

Trigonometrische Substitution anwenden

= \int cos^{4} (u) sin^{3} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (u)) sin (u) cos^{4} (u) d u

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{3 x - 1}{( x + 2 ) ^{2}} d x

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\int 5 x + \frac{2}{x} d x