integral of (cos^5(α))/(sqrt(sin(α)))

Lösung

\int \frac{cos ^{5} ( α )}{sin ( α )} d α

2 sin (α) - \frac{4}{5} sin^{\frac{5}{2}} (α) + \frac{2}{9} sin^{\frac{9}{2}} (α) + C

Schritte zur Lösung

\int \frac{cos ^{5} ( α )}{sin ( α )} d α

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( 1 - sin ^{2} ( α ) ) ^{2} cos ( α )}{sin ( α )} d α

Wende U-Substitution an

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

Multipliziere aus

\frac{( 1 - u ^{2} ) ^{2}}{u} : \frac{1}{u} - 2 u^{\frac{3}{2}} + u^{\frac{7}{2}}

= \int \frac{1}{u} - 2 u^{\frac{3}{2}} + u^{\frac{7}{2}} 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 \frac{1}{u} d u - \int 2 u^{\frac{3}{2}} d u + \int u^{\frac{7}{2}} d u

\int \frac{1}{u} d u = 2 u

\int 2 u^{\frac{3}{2}} d u = \frac{4}{5} u^{\frac{5}{2}}

\int u^{\frac{7}{2}} d u = \frac{2}{9} u^{\frac{9}{2}}

= 2 u - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{9} u^{\frac{9}{2}}

Setze in

u = sin (α)

ein

= 2 sin (α) - \frac{4}{5} sin^{\frac{5}{2}} (α) + \frac{2}{9} sin^{\frac{9}{2}} (α)

Füge eine Konstante zur Lösung hinzu

= 2 sin (α) - \frac{4}{5} sin^{\frac{5}{2}} (α) + \frac{2}{9} sin^{\frac{9}{2}} (α) + C

\frac{d}{d x} (3 sin (x))

x \to 1 lim (e^{2 x})

a re a f (x) = - 4 x^{2} - 4, g (x) = 2 x^{2} - 9

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

x \to 0 lim (\frac{ln ( x )}{3})