integral from pi to 2pi of sin^3(x)

Lösung

\int_{π}^{2 π} sin^{3} (x) d x

- \frac{4}{3}

Dezimale

- 1.33333 \dots

Schritte zur Lösung

\int_{π}^{2 π} sin^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{π}^{2 π} (1 - cos^{2} (x)) sin (x) d x

Wende U-Substitution an

= \int_{- 1}^{1} - 1 + u^{2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{- 1}^{1} 1 d u + \int_{- 1}^{1} u^{2} d u

\int_{- 1}^{1} 1 d u = 2

\int_{- 1}^{1} u^{2} d u = \frac{2}{3}

= - 2 + \frac{2}{3}

Vereinfache

- 2 + \frac{2}{3} : - \frac{4}{3}

= - \frac{4}{3}

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