integral of cos^5(9-11x)

Lösung

\int cos^{5} (9 - 11 x) d x

- \frac{1}{11} (sin (9 - 11 x) - \frac{2 sin ^{3} ( 9 - 11 x )}{3} + \frac{sin ^{5} ( 9 - 11 x )}{5}) + C

Schritte zur Lösung

\int cos^{5} (9 - 11 x) d x

Wende U-Substitution an

= \int - \frac{1}{11} cos^{5} (u) d u

Entferne die Konstante:

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

= - \frac{1}{11} \cdot \int cos^{5} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= - \frac{1}{11} \cdot \int (1 - sin^{2} (u))^{2} cos (u) d u

Wende U-Substitution an

= - \frac{1}{11} \cdot \int (1 - v^{2})^{2} d v

Multipliziere aus

(1 - v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= - \frac{1}{11} \cdot \int 1 - 2 v^{2} + v^{4} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{11} (\int 1 d v - \int 2 v^{2} d v + \int v^{4} d v)

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= - \frac{1}{11} (v - \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

Ersetze zurück

= - \frac{1}{11} (sin (9 - 11 x) - \frac{2 sin ^{3} ( 9 - 11 x )}{3} + \frac{sin ^{5} ( 9 - 11 x )}{5})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{11} (sin (9 - 11 x) - \frac{2 sin ^{3} ( 9 - 11 x )}{3} + \frac{sin ^{5} ( 9 - 11 x )}{5}) + C

\frac{d}{d x} (\frac{3 x - 4}{2 x - 8})

t^{2} y^{^{''}} - 3 t y^{^{'}} + 4 y = t^{2} \cdot ln (t)

x \to 0 lim (x^{6} ln (x))

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

d er i v a t i v e f (x) = \frac{3 x}{x}, x = 8