integral from 0 to pi/4 of cos^3(2x)

Lösung

\int_{0}^{\frac{π}{4}} cos^{3} (2 x) d x

\frac{1}{3}

Dezimale

0.33333 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{4}} cos^{3} (2 x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= \frac{1}{2} \cdot \int_{0}^{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}{2} (\int_{0}^{1} 1 d v - \int_{0}^{1} v^{2} d v)

\int_{0}^{1} 1 d v = 1

\int_{0}^{1} v^{2} d v = \frac{1}{3}

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

Vereinfache

\frac{1}{2} (1 - \frac{1}{3}) : \frac{1}{3}

= \frac{1}{3}

\int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{3} (x) d x

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