integral of sin^{11}(x)cos^6(x)

Lösung

\int sin^{11} (x) cos^{6} (x) d x

- \frac{cos ^{7} ( x )}{7} + \frac{5 cos ^{9} ( x )}{9} - \frac{10 cos ^{11} ( x )}{11} + \frac{10 cos ^{13} ( x )}{13} - \frac{cos ^{15} ( x )}{3} + \frac{cos ^{17} ( x )}{17} + C

Schritte zur Lösung

\int sin^{11} (x) cos^{6} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{5} sin (x) cos^{6} (x) d x

Wende U-Substitution an

= \int - u^{6} (1 - u^{2})^{5} d u

Multipliziere aus

- u^{6} (1 - u^{2})^{5} : - u^{6} + 5 u^{8} - 10 u^{10} + 10 u^{12} - 5 u^{14} + u^{16}

= \int - u^{6} + 5 u^{8} - 10 u^{10} + 10 u^{12} - 5 u^{14} + u^{16} 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^{6} d u + \int 5 u^{8} d u - \int 10 u^{10} d u + \int 10 u^{12} d u - \int 5 u^{14} d u + \int u^{16} d u

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

\int 5 u^{8} d u = \frac{5 u ^{9}}{9}

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

\int 10 u^{12} d u = \frac{10 u ^{13}}{13}

\int 5 u^{14} d u = \frac{u ^{15}}{3}

\int u^{16} d u = \frac{u ^{17}}{17}

= - \frac{u ^{7}}{7} + \frac{5 u ^{9}}{9} - \frac{10 u ^{11}}{11} + \frac{10 u ^{13}}{13} - \frac{u ^{15}}{3} + \frac{u ^{17}}{17}

Setze in

u = cos (x)

ein

= - \frac{cos ^{7} ( x )}{7} + \frac{5 cos ^{9} ( x )}{9} - \frac{10 cos ^{11} ( x )}{11} + \frac{10 cos ^{13} ( x )}{13} - \frac{cos ^{15} ( x )}{3} + \frac{cos ^{17} ( x )}{17}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{7} ( x )}{7} + \frac{5 cos ^{9} ( x )}{9} - \frac{10 cos ^{11} ( x )}{11} + \frac{10 cos ^{13} ( x )}{13} - \frac{cos ^{15} ( x )}{3} + \frac{cos ^{17} ( x )}{17} + C