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

Lösung

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

\frac{4}{3}

Dezimale

1.33333 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= 2 \cdot \int_{0}^{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

= 2 (\int_{0}^{1} 1 d u - \int_{0}^{1} u^{2} d u)

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

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

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

Vereinfache

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

= \frac{4}{3}

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