integral from 0 to pi/3 of sin^3(3x)

Lösung

\int_{0}^{\frac{π}{3}} sin^{3} (3 x) d x

\frac{4}{9}

Dezimale

0.44444 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{3}} sin^{3} (3 x) d x

Wende U-Substitution an

= \int_{0}^{π} sin^{3} (u) \frac{1}{3} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int_{0}^{π} sin^{3} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{3} \cdot \int_{0}^{π} (1 - cos^{2} (u)) sin (u) d u

Wende U-Substitution an

= \frac{1}{3} \cdot \int_{1}^{- 1} - 1 + v^{2} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= \frac{1}{3} (- \int_{- 1}^{1} - 1 + v^{2} d v)

Wende die Summenregel an:

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

= \frac{1}{3} (- (- \int_{- 1}^{1} 1 d v + \int_{- 1}^{1} v^{2} d v))

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

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

= \frac{1}{3} (- (- 2 + \frac{2}{3}))

Vereinfache

\frac{1}{3} (- (- 2 + \frac{2}{3})) : \frac{4}{9}

= \frac{4}{9}

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