integral from 0 to pi/2 of 30cos^5(x)

Lösung

\int_{0}^{\frac{π}{2}} 30 cos^{5} (x) d x

16

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} 30 cos^{5} (x) d x

Entferne die Konstante:

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

= 30 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 30 \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{2} cos (x) d x

Wende U-Substitution an

= 30 \cdot \int_{0}^{1} (1 - u^{2})^{2} d u

Multipliziere aus

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

= 30 \cdot \int_{0}^{1} 1 - 2 u^{2} + u^{4} 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

= 30 (\int_{0}^{1} 1 d u - \int_{0}^{1} 2 u^{2} d u + \int_{0}^{1} u^{4} d u)

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

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

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

= 30 (1 - \frac{2}{3} + \frac{1}{5})

Vereinfache

30 (1 - \frac{2}{3} + \frac{1}{5}) : 16

= 16

t an g e n t f (x) = 3 x^{2} + 2 x, at x = 1

n = 1 \sum \infty (- 0.3)^{n}

x \to 0 lim (\frac{1}{3} x)

t an g e n t f (x) = 3 x - cos (x), at x = 0

\int (9 x + 5)^{2} d x