integral of (sin(x))^5(cos(x))^4

Lösung

\int (sin (x))^{5} (cos (x))^{4} d x

- \frac{cos ^{5} ( x )}{5} + \frac{2 cos ^{7} ( x )}{7} - \frac{cos ^{9} ( x )}{9} + C

Schritte zur Lösung

\int (sin (x))^{5} (cos (x))^{4} d x

Vereinfache

= \int sin^{5} (x) cos^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{4} (x) d x

Wende U-Substitution an

= \int - u^{4} (1 - u^{2})^{2} d u

Multipliziere aus

- u^{4} (1 - u^{2})^{2} : - u^{4} + 2 u^{6} - u^{8}

= \int - u^{4} + 2 u^{6} - u^{8} 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^{4} d u + \int 2 u^{6} d u - \int u^{8} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 2 u^{6} d u = \frac{2 u ^{7}}{7}

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

= - \frac{u ^{5}}{5} + \frac{2 u ^{7}}{7} - \frac{u ^{9}}{9}

Setze in

u = cos (x)

ein

= - \frac{cos ^{5} ( x )}{5} + \frac{2 cos ^{7} ( x )}{7} - \frac{cos ^{9} ( x )}{9}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{5} ( x )}{5} + \frac{2 cos ^{7} ( x )}{7} - \frac{cos ^{9} ( x )}{9} + C

\int 2 x 4 - x d x

\int sec^{5} (5 x) tan (5 x) d x

\int \frac{2}{x} + e^{2 x} d x

\int 8 t e^{7} t d t

\int \frac{sec ( θ ) tan ( θ )}{1 - sec ( θ )} d θ