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Beliebt Rechnen >

integral of sin^8(x)cos^7(x)

Lösung

\int sin^{8} (x) cos^{7} (x) d x

Lösung

\frac{sin ^{9} ( x )}{9} - \frac{3 sin ^{11} ( x )}{11} + \frac{3 sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{15} + C

Schritte zur Lösung

\int sin^{8} (x) cos^{7} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (x))^{3} cos (x) sin^{8} (x) d x

Wende U-Substitution an

= \int u^{8} (1 - u^{2})^{3} d u

Multipliziere aus

u^{8} (1 - u^{2})^{3} : u^{8} - 3 u^{10} + 3 u^{12} - u^{14}

= \int u^{8} - 3 u^{10} + 3 u^{12} - u^{14} 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 u^{8} d u - \int 3 u^{10} d u + \int 3 u^{12} d u - \int u^{14} d u

\int u^{8} d u = \frac{u ^{9}}{9}

\int 3 u^{10} d u = \frac{3 u ^{11}}{11}

\int 3 u^{12} d u = \frac{3 u ^{13}}{13}

\int u^{14} d u = \frac{u ^{15}}{15}

= \frac{u ^{9}}{9} - \frac{3 u ^{11}}{11} + \frac{3 u ^{13}}{13} - \frac{u ^{15}}{15}

Setze in

u = sin (x)

ein

= \frac{sin ^{9} ( x )}{9} - \frac{3 sin ^{11} ( x )}{11} + \frac{3 sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{15}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{9} ( x )}{9} - \frac{3 sin ^{11} ( x )}{11} + \frac{3 sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{15} + C

Graph

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