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Beliebt Rechnen >

integral of sin^5(x)cos^9(x)

Lösung

\int sin^{5} (x) cos^{9} (x) d x

Lösung

- \frac{cos ^{10} ( x )}{10} + \frac{cos ^{12} ( x )}{6} - \frac{cos ^{14} ( x )}{14} + C

Schritte zur Lösung

\int sin^{5} (x) cos^{9} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{9} (x) d x

Wende U-Substitution an

= \int - u^{9} (1 - u^{2})^{2} d u

Multipliziere aus

- u^{9} (1 - u^{2})^{2} : - u^{9} + 2 u^{11} - u^{13}

= \int - u^{9} + 2 u^{11} - u^{13} 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^{9} d u + \int 2 u^{11} d u - \int u^{13} d u

\int u^{9} d u = \frac{u ^{10}}{10}

\int 2 u^{11} d u = \frac{u ^{12}}{6}

\int u^{13} d u = \frac{u ^{14}}{14}

= - \frac{u ^{10}}{10} + \frac{u ^{12}}{6} - \frac{u ^{14}}{14}

Setze in

u = cos (x)

ein

= - \frac{cos ^{10} ( x )}{10} + \frac{cos ^{12} ( x )}{6} - \frac{cos ^{14} ( x )}{14}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{10} ( x )}{10} + \frac{cos ^{12} ( x )}{6} - \frac{cos ^{14} ( x )}{14} + C

Graph

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