integral from 0 to pi/2 of cos^7(x)

Lösung

\int_{0}^{\frac{π}{2}} cos^{7} (x) d x

\frac{16}{35}

Dezimale

0.45714 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} cos^{7} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= \int_{0}^{1} (1 - u^{2})^{3} d u

Multipliziere aus

(1 - u^{2})^{3} : 1 - 3 u^{2} + 3 u^{4} - u^{6}

= \int_{0}^{1} 1 - 3 u^{2} + 3 u^{4} - u^{6} 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

= \int_{0}^{1} 1 d u - \int_{0}^{1} 3 u^{2} d u + \int_{0}^{1} 3 u^{4} d u - \int_{0}^{1} u^{6} d u

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

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

\int_{0}^{1} 3 u^{4} d u = \frac{3}{5}

\int_{0}^{1} u^{6} d u = \frac{1}{7}

= 1 - 1 + \frac{3}{5} - \frac{1}{7}

Vereinfache

1 - 1 + \frac{3}{5} - \frac{1}{7} : \frac{16}{35}

= \frac{16}{35}

6 y^{^{''}} + 10 y^{^{'}} = 0

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