integral of (cos^7(x))/(sqrt(sin(x)))

Lösung

\int \frac{cos ^{7} ( x )}{sin ( x )} d x

2 sin (x) - \frac{6}{5} sin^{\frac{5}{2}} (x) + \frac{2}{3} sin^{\frac{9}{2}} (x) - \frac{2}{13} sin^{\frac{13}{2}} (x) + C

Schritte zur Lösung

\int \frac{cos ^{7} ( x )}{sin ( x )} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( 1 - sin ^{2} ( x ) ) ^{3} cos ( x )}{sin ( x )} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{( 1 - u ^{2} ) ^{3}}{u} : \frac{1}{u} - 3 u^{\frac{3}{2}} + 3 u^{\frac{7}{2}} - u^{\frac{11}{2}}

= \int \frac{1}{u} - 3 u^{\frac{3}{2}} + 3 u^{\frac{7}{2}} - u^{\frac{11}{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 3 u^{\frac{3}{2}} d u + \int 3 u^{\frac{7}{2}} d u - \int u^{\frac{11}{2}} d u

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

\int 3 u^{\frac{3}{2}} d u = \frac{6}{5} u^{\frac{5}{2}}

\int 3 u^{\frac{7}{2}} d u = \frac{2}{3} u^{\frac{9}{2}}

\int u^{\frac{11}{2}} d u = \frac{2}{13} u^{\frac{13}{2}}

= 2 u - \frac{6}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{9}{2}} - \frac{2}{13} u^{\frac{13}{2}}

Setze in

u = sin (x)

ein

= 2 sin (x) - \frac{6}{5} sin^{\frac{5}{2}} (x) + \frac{2}{3} sin^{\frac{9}{2}} (x) - \frac{2}{13} sin^{\frac{13}{2}} (x)

Füge eine Konstante zur Lösung hinzu

= 2 sin (x) - \frac{6}{5} sin^{\frac{5}{2}} (x) + \frac{2}{3} sin^{\frac{9}{2}} (x) - \frac{2}{13} sin^{\frac{13}{2}} (x) + C

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