integral of (16-x^2)^{3/2}

Lösung

\int (16 - x^{2})^{\frac{3}{2}} d x

x (16 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 16)^{\frac{1}{2}} + 32 arcsin (\frac{1}{4} x) - 16 sin (2 arcsin (\frac{1}{4} x))) + C

Schritte zur Lösung

\int (16 - x^{2})^{\frac{3}{2}} d x

Wende die partielle Integration an

= x (16 - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} (16 - x^{2})^{\frac{1}{2}} d x

\int - 3 x^{2} (16 - x^{2})^{\frac{1}{2}} d x = - 3 (\frac{1}{4} x^{3} (- x^{2} + 16)^{\frac{1}{2}} + 32 arcsin (\frac{1}{4} x) - 16 sin (2 arcsin (\frac{1}{4} x)))

= x (16 - x^{2})^{\frac{3}{2}} - (- 3 (\frac{1}{4} x^{3} (- x^{2} + 16)^{\frac{1}{2}} + 32 arcsin (\frac{1}{4} x) - 16 sin (2 arcsin (\frac{1}{4} x))))

Vereinfache

= x (16 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 16)^{\frac{1}{2}} + 32 arcsin (\frac{1}{4} x) - 16 sin (2 arcsin (\frac{1}{4} x)))

Füge eine Konstante zur Lösung hinzu

= x (16 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 16)^{\frac{1}{2}} + 32 arcsin (\frac{1}{4} x) - 16 sin (2 arcsin (\frac{1}{4} x))) + C

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