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Beliebt Rechnen >

integral of (cos^5(a))/(sqrt(sin(a)))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

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 (a)

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-1)/(4x^2+4x+5)

\int \frac{x - 1}{4 x ^{2} + 4 x + 5} d x

integral of 1/(sqrt(5x^2-x-1))

\int \frac{1}{5 x ^{2} - x - 1} d x

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

integral of sec(9x)tan(9x)

\int sec (9 x) tan (9 x) d x