integral of (cot^5(x))/(sqrt(sin(x)))

Lösung

\int \frac{cot ^{5} ( x )}{sin ( x )} d x

- \frac{2}{9} csc^{\frac{9}{2}} (x) + \frac{4}{5} csc^{\frac{5}{2}} (x) - 2 csc (x) + C

Schritte zur Lösung

\int \frac{cot ^{5} ( x )}{sin ( x )} d x

Wende U-Substitution an

= \int \frac{( 1 - u ^{2} ) ^{2}}{u ^{5} u} d u

Multipliziere aus

\frac{( 1 - u ^{2} ) ^{2}}{u ^{5} u} : \frac{1}{u ^{5} u} - \frac{2}{u ^{3} u} + \frac{1}{u u}

= \int \frac{1}{u ^{5} u} - \frac{2}{u ^{3} u} + \frac{1}{u u} 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 ^{5} u} d u - \int \frac{2}{u ^{3} u} d u + \int \frac{1}{u u} d u

\int \frac{1}{u ^{5} u} d u = - \frac{2}{9 u ^{\frac{9}{2}}}

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

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

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

Setze in

u = sin (x)

ein

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

Vereinfache

- \frac{2}{9 sin ^{\frac{9}{2}} ( x )} - (- \frac{4}{5 sin ^{\frac{5}{2}} ( x )}) - \frac{2}{sin ( x )} : - \frac{2}{9} csc^{\frac{9}{2}} (x) + \frac{4}{5} csc^{\frac{5}{2}} (x) - 2 csc (x)

= - \frac{2}{9} csc^{\frac{9}{2}} (x) + \frac{4}{5} csc^{\frac{5}{2}} (x) - 2 csc (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{9} csc^{\frac{9}{2}} (x) + \frac{4}{5} csc^{\frac{5}{2}} (x) - 2 csc (x) + C

x \to - 3 lim (x)

\int \frac{sin ( x ) + cos ( x )}{sin ^{2} ( x )} d x

\int a rco sin (x) d x

t an g e n t x^{2} - 3

f (x) = x^{2} - cos (2 x)