What is the integral of (cot^5(x))/(sqrt(sin(x))) ?

The integral of (cot^5(x))/(sqrt(sin(x))) is -2/9 csc^{9/2}(x)+4/5 csc^{5/2}(x)-2sqrt(csc(x))+C

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

\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

Solution steps

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

Apply u-substitution

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

Expand

\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

Apply the Sum Rule:

\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}

Substitute back

u = sin (x)

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

Simplify

- \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)

Add a constant to the solution

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

What is the integral of (cot^5(x))/(sqrt(sin(x))) ?
The integral of (cot^5(x))/(sqrt(sin(x))) is -2/9 csc^{9/2}(x)+4/5 csc^{5/2}(x)-2sqrt(csc(x))+C