integral of (sin(x))^8(cos(x))^3

Lösung

\int (sin (x))^{8} (cos (x))^{3} d x

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

Schritte zur Lösung

\int (sin (x))^{8} (cos (x))^{3} d x

Vereinfache

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

u^{8} (1 - u^{2}) : u^{8} - u^{10}

= \int u^{8} - u^{10} 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 u^{10} d u

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

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

= \frac{u ^{9}}{9} - \frac{u ^{11}}{11}

Setze in

u = sin (x)

ein

= \frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11}

Füge eine Konstante zur Lösung hinzu

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

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