integral of (sin(x))^6(cos(x))^5

Lösung

\int (sin (x))^{6} (cos (x))^{5} d x

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

Schritte zur Lösung

\int (sin (x))^{6} (cos (x))^{5} d x

Vereinfache

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

= \frac{u ^{7}}{7} - \frac{2 u ^{9}}{9} + \frac{u ^{11}}{11}

Setze in

u = sin (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{8}{9 x 81 x ^{2} - 64} d x

\int cos (x^{2}) x^{3} d x

\int \frac{x ^{2} + 3 x + 2}{x ^{3} + 4 x} d x

\int - 5 x d x

\int \frac{e ^{2 x}}{7 - e ^{4 x}} d x