integral of sin^2(x)cos^9(x)

Lösung

\int sin^{2} (x) cos^{9} (x) d x

\frac{sin ^{3} ( x )}{3} - \frac{4 sin ^{5} ( x )}{5} + \frac{6 sin ^{7} ( x )}{7} - \frac{4 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

Schritte zur Lösung

\int sin^{2} (x) cos^{9} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (x))^{4} cos (x) sin^{2} (x) d x

Wende U-Substitution an

= \int u^{2} (1 - u^{2})^{4} d u

Multipliziere aus

u^{2} (1 - u^{2})^{4} : u^{2} - 4 u^{4} + 6 u^{6} - 4 u^{8} + u^{10}

= \int u^{2} - 4 u^{4} + 6 u^{6} - 4 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^{2} d u - \int 4 u^{4} d u + \int 6 u^{6} d u - \int 4 u^{8} d u + \int u^{10} d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 4 u^{4} d u = \frac{4 u ^{5}}{5}

\int 6 u^{6} d u = \frac{6 u ^{7}}{7}

\int 4 u^{8} d u = \frac{4 u ^{9}}{9}

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

= \frac{u ^{3}}{3} - \frac{4 u ^{5}}{5} + \frac{6 u ^{7}}{7} - \frac{4 u ^{9}}{9} + \frac{u ^{11}}{11}

Setze in

u = sin (x)

ein

= \frac{sin ^{3} ( x )}{3} - \frac{4 sin ^{5} ( x )}{5} + \frac{6 sin ^{7} ( x )}{7} - \frac{4 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{3} ( x )}{3} - \frac{4 sin ^{5} ( x )}{5} + \frac{6 sin ^{7} ( x )}{7} - \frac{4 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

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