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

Lösung

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

x (a^{2} - x^{2})^{\frac{3}{2}} + \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a}))) + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int - 3 x^{2} (a^{2} - x^{2})^{\frac{1}{2}} d x = - \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a})))

= x (a^{2} - x^{2})^{\frac{3}{2}} - (- \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a}))))

Vereinfache

= x (a^{2} - x^{2})^{\frac{3}{2}} + \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a})))

Füge eine Konstante zur Lösung hinzu

= x (a^{2} - x^{2})^{\frac{3}{2}} + \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a}))) + C

d er i v a t i v e lo g_{5} (x^{2} - 4)

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