integral from-2 to 2 of (x-3)sqrt(4-x^2)

Lösung

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

- 6 π

Dezimale

- 18.84955 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

[(x - 3) (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - (x (\frac{1}{2} x 4 - x^{2} + 2 arcsin (\frac{x}{2})) + \frac{1}{3} (- x^{2} + 4)^{\frac{3}{2}})]_{- 2}^{2} : [\frac{1}{6} (- 3 x^{2} - x^{2} + 4 + 6 x sin (2 arcsin (\frac{x}{2})) - 2 (- x^{2} + 4)^{\frac{3}{2}} - 36 arcsin (\frac{x}{2}) - 18 sin (2 arcsin (\frac{x}{2})))]_{- 2}^{2}

= [\frac{1}{6} (- 3 x^{2} - x^{2} + 4 + 6 x sin (2 arcsin (\frac{x}{2})) - 2 (- x^{2} + 4)^{\frac{3}{2}} - 36 arcsin (\frac{x}{2}) - 18 sin (2 arcsin (\frac{x}{2})))]_{- 2}^{2}

Berechne die Grenzen:

- 6 π

= - 6 π

\int_{3}^{5} x ln (x) d x

\int_{2}^{6} 2 x^{2} - 16 x + 24 d x

\int_{0}^{5} \frac{1}{x} d x

\int_{\frac{π}{4}}^{\frac{3 π}{4}} cot (x) d x

\int_{π}^{2 π} 5 sin (2 x) d x