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

Lösung

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

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

ein

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d y}{d x} 0.4 x y = 9 x

\int_{2}^{10} \frac{x ^{2} + 36}{12 x} d x

x^{^{'''}} - x^{^{''}} - x^{^{'}} + x = 2 e^{- t} + 3

\frac{\partial}{\partial x} (8 x - 2 y - 2)

\frac{\partial}{\partial y} (\frac{y}{z})