integral from 0 to pi of xsin^5(x)

Lösung

\int_{0}^{π} x sin^{5} (x) d x

\frac{8 π}{15}

Dezimale

1.67551 \dots

Schritte zur Lösung

\int_{0}^{π} x sin^{5} (x) d x

Wende die partielle Integration an

= [x (- cos (x) + \frac{2}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)) - \int - cos (x) + \frac{2}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x) d x]_{0}^{π}

\int - cos (x) + \frac{2}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x) d x = - \frac{8}{15} sin (x) - \frac{2}{9} sin^{3} (x) + \frac{2}{15} sin^{3} (x) - \frac{1}{25} sin^{5} (x)

= [x (- cos (x) + \frac{2}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)) - (- \frac{8}{15} sin (x) - \frac{2}{9} sin^{3} (x) + \frac{2}{15} sin^{3} (x) - \frac{1}{25} sin^{5} (x))]_{0}^{π}

Vereinfache

[x (- cos (x) + \frac{2}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)) - (- \frac{8}{15} sin (x) - \frac{2}{9} sin^{3} (x) + \frac{2}{15} sin^{3} (x) - \frac{1}{25} sin^{5} (x))]_{0}^{π} : [\frac{1}{225} (- 225 x cos (x) + 150 x cos^{3} (x) - 45 x cos^{5} (x) + 120 sin (x) + 20 sin^{3} (x) + 9 sin^{5} (x))]_{0}^{π}

= [\frac{1}{225} (- 225 x cos (x) + 150 x cos^{3} (x) - 45 x cos^{5} (x) + 120 sin (x) + 20 sin^{3} (x) + 9 sin^{5} (x))]_{0}^{π}

Berechne die Grenzen:

\frac{8 π}{15}

= \frac{8 π}{15}