integral of (sin(x))^5(cos(x))^3

Lösung

\int (sin (x))^{5} (cos (x))^{3} d x

- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

Schritte zur Lösung

\int (sin (x))^{5} (cos (x))^{3} d x

Vereinfache

= \int sin^{5} (x) cos^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{3} (x) d x

Wende U-Substitution an

= \int - u^{3} (1 - u^{2})^{2} d u

Multipliziere aus

- u^{3} (1 - u^{2})^{2} : - u^{3} + 2 u^{5} - u^{7}

= \int - u^{3} + 2 u^{5} - u^{7} 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^{3} d u + \int 2 u^{5} d u - \int u^{7} d u

\int u^{3} d u = \frac{u ^{4}}{4}

\int 2 u^{5} d u = \frac{u ^{6}}{3}

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

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{3} - \frac{u ^{8}}{8}

Setze in

u = cos (x)

ein

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

\frac{d ^{2}}{d x ^{2}} (1 + \frac{1}{x ^{4}})

d er i v a t i v e \frac{3}{x}

(\frac{d x}{d y}) + 3 x y^{3} = 0

d er i v a t i v e x + 19

d er i v a t i v e t^{\frac{1}{3}}