integral from 0 to pi/8 of 3cos^5(4x)

Lösung

\int_{0}^{\frac{π}{8}} 3 cos^{5} (4 x) d x

\frac{2}{5}

Dezimale

0.4

Schritte zur Lösung

\int_{0}^{\frac{π}{8}} 3 cos^{5} (4 x) d x

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{\frac{π}{8}} cos^{5} (4 x) d x

Wende U-Substitution an

= 3 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (u) \frac{1}{4} d u

Entferne die Konstante:

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

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} cos^{5} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (u))^{2} cos (u) d u

Wende U-Substitution an

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{1} (1 - v^{2})^{2} d v

Multipliziere aus

(1 - v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{1} 1 - 2 v^{2} + v^{4} 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

= 3 \cdot \frac{1}{4} (\int_{0}^{1} 1 d v - \int_{0}^{1} 2 v^{2} d v + \int_{0}^{1} v^{4} d v)

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

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

\int_{0}^{1} v^{4} d v = \frac{1}{5}

= 3 \cdot \frac{1}{4} (1 - \frac{2}{3} + \frac{1}{5})

Vereinfache

3 \cdot \frac{1}{4} (1 - \frac{2}{3} + \frac{1}{5}) : \frac{2}{5}

= \frac{2}{5}