integral of 2cos^5(x)sin^4(x)

Lösung

\int 2 cos^{5} (x) sin^{4} (x) d x

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

Schritte zur Lösung

\int 2 cos^{5} (x) sin^{4} (x) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos^{5} (x) sin^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= 2 \cdot \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

= 2 (\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}

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

Setze in

u = sin (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

t an g e n t f (x) = - \frac{9 x ^{2}}{2} + 8 x - 22

\frac{d h}{d t} = - h^{\frac{1}{3}}

\int (x^{2})^{23} d (x^{2})^{2}

d er i v a t i v e x e^{e}

d er i v a t i v e y = - 2 x