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Beliebt Rechnen >

integral of cos^9(x)

Lösung

\int cos^{9} (x) d x

Lösung

sin (x) - \frac{4 sin ^{3} ( x )}{3} + \frac{6 sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9} + C

Schritte zur Lösung

\int cos^{9} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int 1 - 4 u^{2} + 6 u^{4} - 4 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 1 d u - \int 4 u^{2} d u + \int 6 u^{4} d u - \int 4 u^{6} d u + \int u^{8} d u

\int 1 d u = u

\int 4 u^{2} d u = \frac{4 u ^{3}}{3}

\int 6 u^{4} d u = \frac{6 u ^{5}}{5}

\int 4 u^{6} d u = \frac{4 u ^{7}}{7}

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

= u - \frac{4 u ^{3}}{3} + \frac{6 u ^{5}}{5} - \frac{4 u ^{7}}{7} + \frac{u ^{9}}{9}

Setze in

u = sin (x)

ein

= sin (x) - \frac{4 sin ^{3} ( x )}{3} + \frac{6 sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9}

Füge eine Konstante zur Lösung hinzu

= sin (x) - \frac{4 sin ^{3} ( x )}{3} + \frac{6 sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9} + C

Graph

Beliebte Beispiele

integral of (8x^3-3x^2+8)

\int (8 x^{3} - 3 x^{2} + 8) d x

derivative of x^5e^{6x}

\frac{d}{d x} (x^{5} e^{6 x})

limit as x approaches infinity of 2/x

x \to \infty lim (\frac{2}{x})

derivative of-2cos(3x)

\frac{d}{d x} (- 2 cos (3 x))

limit as x approaches 0+of x/(3x)-7x

x \to 0 + lim (\frac{x}{3 x} - 7 x)