integral from 0 to 2 of (4-x^2)^{3/2}

Lösung

\int_{0}^{2} (4 - x^{2})^{\frac{3}{2}} d x

3 π

Dezimale

9.42477 \dots

Schritte zur Lösung

\int_{0}^{2} (4 - x^{2})^{\frac{3}{2}} d x

Wende die partielle Integration an

= [x (4 - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} (4 - x^{2})^{\frac{1}{2}} d x]_{0}^{2}

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

= [x (4 - x^{2})^{\frac{3}{2}} - (- 3 (\frac{1}{4} x^{3} (- x^{2} + 4)^{\frac{1}{2}} + 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x))))]_{0}^{2}

Vereinfache

= [x (4 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 4)^{\frac{1}{2}} + 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x)))]_{0}^{2}

Berechne die Grenzen:

3 π

= 3 π

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