integral of sin^4(x)cos^9(x)

Lösung

\int sin^{4} (x) cos^{9} (x) d x

\frac{sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{2 sin ^{9} ( x )}{3} - \frac{4 sin ^{11} ( x )}{11} + \frac{sin ^{13} ( x )}{13} + C

Schritte zur Lösung

\int sin^{4} (x) cos^{9} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

u^{4} (1 - u^{2})^{4} : u^{4} - 4 u^{6} + 6 u^{8} - 4 u^{10} + u^{12}

= \int u^{4} - 4 u^{6} + 6 u^{8} - 4 u^{10} + u^{12} 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^{4} d u - \int 4 u^{6} d u + \int 6 u^{8} d u - \int 4 u^{10} d u + \int u^{12} d u

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

\int 4 u^{6} d u = \frac{4 u ^{7}}{7}

\int 6 u^{8} d u = \frac{2 u ^{9}}{3}

\int 4 u^{10} d u = \frac{4 u ^{11}}{11}

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

= \frac{u ^{5}}{5} - \frac{4 u ^{7}}{7} + \frac{2 u ^{9}}{3} - \frac{4 u ^{11}}{11} + \frac{u ^{13}}{13}

Setze in

u = sin (x)

ein

= \frac{sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{2 sin ^{9} ( x )}{3} - \frac{4 sin ^{11} ( x )}{11} + \frac{sin ^{13} ( x )}{13}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{2 sin ^{9} ( x )}{3} - \frac{4 sin ^{11} ( x )}{11} + \frac{sin ^{13} ( x )}{13} + C

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