integral of (sin(x))^{11}(cos(x))^5

Lösung

\int (sin (x))^{11} (cos (x))^{5} d x

- \frac{cos ^{6} ( x )}{6} + \frac{5 cos ^{8} ( x )}{8} - cos^{10} (x) + \frac{5 cos ^{12} ( x )}{6} - \frac{5 cos ^{14} ( x )}{14} + \frac{cos ^{16} ( x )}{16} + C

Schritte zur Lösung

\int (sin (x))^{11} (cos (x))^{5} d x

Vereinfache

= \int sin^{11} (x) cos^{5} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

- u^{5} (1 - u^{2})^{5} : - u^{5} + 5 u^{7} - 10 u^{9} + 10 u^{11} - 5 u^{13} + u^{15}

= \int - u^{5} + 5 u^{7} - 10 u^{9} + 10 u^{11} - 5 u^{13} + u^{15} 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^{5} d u + \int 5 u^{7} d u - \int 10 u^{9} d u + \int 10 u^{11} d u - \int 5 u^{13} d u + \int u^{15} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int 5 u^{7} d u = \frac{5 u ^{8}}{8}

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

\int 10 u^{11} d u = \frac{5 u ^{12}}{6}

\int 5 u^{13} d u = \frac{5 u ^{14}}{14}

\int u^{15} d u = \frac{u ^{16}}{16}

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

Setze in

u = cos (x)

ein

= - \frac{cos ^{6} ( x )}{6} + \frac{5 cos ^{8} ( x )}{8} - cos^{10} (x) + \frac{5 cos ^{12} ( x )}{6} - \frac{5 cos ^{14} ( x )}{14} + \frac{cos ^{16} ( x )}{16}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{6} ( x )}{6} + \frac{5 cos ^{8} ( x )}{8} - cos^{10} (x) + \frac{5 cos ^{12} ( x )}{6} - \frac{5 cos ^{14} ( x )}{14} + \frac{cos ^{16} ( x )}{16} + C

y^{^{'}} = y^{3} - 6 x^{2} y^{3}, y (1) = - \frac{1}{2}

x \to 1 lim (\frac{x ^{2} + 1}{x + 1})

d er i v a t i v e f (x) = 80640 (2 x + 5)^{2}

t an g e n t f (x) = x^{2} - x

d er i v a t i v e 2 x x