integral (cos^3(x))/(sqrt(sin(x)))

Lösung

integral

\frac{cos ^{3} ( x )}{sin ( x )}

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

Schritte zur Lösung

\int \frac{cos ^{3} ( x )}{sin ( x )} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

\frac{1 - u ^{2}}{u} : \frac{1}{u} - u^{\frac{3}{2}}

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

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

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

= 2 u - \frac{2}{5} u^{\frac{5}{2}}

Setze in

u = sin (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{( x ^{2} )}{( ( 2 x + x ^{3} ) ^{3} )}

in t e g r a l v = \frac{1}{3 t + 2}

in t e g r a l \frac{( x + 1 )}{( x ^{2} + x + 1 )}

in t e g r a l ln (\frac{e ^{x}}{x ^{2}})

in t e g r a l (x^{2} - a^{2})^{\frac{1}{2}}