integral from-3 to 3 of (x-3)sqrt(9-x^2)

Lösung

\int_{- 3}^{3} (x - 3) 9 - x^{2} d x

- \frac{27 π}{2}

Dezimale

- 42.41150 \dots

Schritte zur Lösung

\int_{- 3}^{3} (x - 3) 9 - x^{2} d x

Wende die partielle Integration an

= [\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \int \frac{9}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) d x]_{- 3}^{3}

\int \frac{9}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) d x = \frac{9}{2} (x arcsin (\frac{x}{3}) + 9 - x^{2} - \frac{1}{27} (9 - x^{2})^{\frac{3}{2}})

= [\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \frac{9}{2} (x arcsin (\frac{x}{3}) + 9 - x^{2} - \frac{1}{27} (9 - x^{2})^{\frac{3}{2}})]_{- 3}^{3}

Vereinfache

[\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \frac{9}{2} (x arcsin (\frac{x}{3}) + 9 - x^{2} - \frac{1}{27} (9 - x^{2})^{\frac{3}{2}})]_{- 3}^{3} : [\frac{1}{12} (27 x sin (2 arcsin (\frac{x}{3})) - 162 arcsin (\frac{x}{3}) - 81 sin (2 arcsin (\frac{x}{3})) - 54 - x^{2} + 9 + 2 (- x^{2} + 9)^{\frac{3}{2}})]_{- 3}^{3}

= [\frac{1}{12} (27 x sin (2 arcsin (\frac{x}{3})) - 162 arcsin (\frac{x}{3}) - 81 sin (2 arcsin (\frac{x}{3})) - 54 - x^{2} + 9 + 2 (- x^{2} + 9)^{\frac{3}{2}})]_{- 3}^{3}

Berechne die Grenzen:

- \frac{27 π}{2}

= - \frac{27 π}{2}

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