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Beliebt Rechnen >

integral from 0 to pi/4 of cos^7(x)

Lösung

\int_{0}^{\frac{π}{4}} cos^{7} (x) d x

Lösung

\frac{177}{280 2}

+1

Dezimale

0.44699 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{4}} cos^{7} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

\int_{0}^{\frac{2}{2}} 3 u^{4} d u = \frac{3}{20 2}

\int_{0}^{\frac{2}{2}} u^{6} d u = \frac{1}{56 2}

= \frac{1}{2} - \frac{1}{2 2} + \frac{3}{20 2} - \frac{1}{56 2}

Vereinfache

\frac{1}{2} - \frac{1}{2 2} + \frac{3}{20 2} - \frac{1}{56 2} : \frac{177}{280 2}

= \frac{177}{280 2}

Graph

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