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Beliebt Rechnen >

integral of sqrt((a^2-x^2)^3)

Lösung

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

Lösung

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})^{3} d x

(a^{2} - x^{2})^{3} = (a^{2} - x^{2})^{\frac{3}{2}},

angenommen

(a^{2} - x^{2}) \geq 0

= \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} d x

\int - 3 x^{2} a^{2} - x^{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

Graph

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