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

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

ein

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{cos ( x ) + sin ( x )}{cos ( x )} d x

\int \frac{x + 2}{x x + 1} d x

\int \frac{y ^{2}}{( 9 - y ^{2} ) ^{\frac{3}{2}}} d y

\int \frac{2 x + 1}{( x ^{2} + x ) ^{3}} d x

\int \frac{1}{( x ^{2} + 3 x + 2 )} d x