integral of 9cos^5(x)sin^4(x)

Lösung

\int 9 cos^{5} (x) sin^{4} (x) d x

\frac{9}{5} sin^{5} (x) - \frac{18}{7} sin^{7} (x) + sin^{9} (x) + C

Schritte zur Lösung

\int 9 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

= 9 \cdot \int cos^{5} (x) sin^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 9 \cdot \int (1 - sin^{2} (x))^{2} cos (x) sin^{4} (x) d x

Wende U-Substitution an

= 9 \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}

= 9 \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

= 9 (\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}

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

Setze in

u = sin (x)

ein

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

Vereinfache

9 (\frac{sin ^{5} ( x )}{5} - \frac{2 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9}) : \frac{9}{5} sin^{5} (x) - \frac{18}{7} sin^{7} (x) + sin^{9} (x)

= \frac{9}{5} sin^{5} (x) - \frac{18}{7} sin^{7} (x) + sin^{9} (x)

Füge eine Konstante zur Lösung hinzu

= \frac{9}{5} sin^{5} (x) - \frac{18}{7} sin^{7} (x) + sin^{9} (x) + C

\frac{\partial}{\partial x} (2 x y + 3 x - 4 y + 1)

\frac{\partial}{\partial x} (x y^{7} - x^{2} y)

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

d er i v a t i v e f (x) = \frac{2 - x ^{2}}{3 + x ^{2}}

n = 1 \sum \infty arctan (18 n)