integral from 0 to pi of xsin^3(x)

Lösung

\int_{0}^{π} x sin^{3} (x) d x

\frac{2 π}{3}

Dezimale

2.09439 \dots

Schritte zur Lösung

\int_{0}^{π} x sin^{3} (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

[x (- cos (x) + \frac{1}{3} cos^{3} (x)) - (- \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x))]_{0}^{π} : [\frac{1}{9} (- 9 x cos (x) + 3 x cos^{3} (x) + 6 sin (x) + sin^{3} (x))]_{0}^{π}

= [\frac{1}{9} (- 9 x cos (x) + 3 x cos^{3} (x) + 6 sin (x) + sin^{3} (x))]_{0}^{π}

Berechne die Grenzen:

\frac{2 π}{3}

= \frac{2 π}{3}

\int_{- 1}^{4} \frac{3}{4 x ^{2}} d x

\int_{0}^{1} (x^{2} + 5) e^{2 x} d x

\int_{\frac{π}{4}}^{2 π} cos (x) d x

\int_{0}^{2} 5 (1 - e^{y}) d y

\int_{0}^{1} \frac{ln ( x )}{x ^{\frac{1}{3}}} d x