What is the integral of csc^5(x)cot^5(x) ?

The integral of csc^5(x)cot^5(x) is -1/9 csc^9(x)+2/7 csc^7(x)-1/5 csc^5(x)+C

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integral of csc^5(x)cot^5(x)

\int csc^{5} (x) cot^{5} (x) d x

- \frac{1}{9} csc^{9} (x) + \frac{2}{7} csc^{7} (x) - \frac{1}{5} csc^{5} (x) + C

Solution steps

\int csc^{5} (x) cot^{5} (x) d x

Apply u-substitution

= \int \frac{( 1 - u ^{2} ) ^{2}}{u ^{10}} d u

Expand

\frac{( 1 - u ^{2} ) ^{2}}{u ^{10}} : \frac{1}{u ^{10}} - \frac{2}{u ^{8}} + \frac{1}{u ^{6}}

= \int \frac{1}{u ^{10}} - \frac{2}{u ^{8}} + \frac{1}{u ^{6}} 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 ^{10}} d u - \int \frac{2}{u ^{8}} d u + \int \frac{1}{u ^{6}} d u

\int \frac{1}{u ^{10}} d u = - \frac{1}{9 u ^{9}}

\int \frac{2}{u ^{8}} d u = - \frac{2}{7 u ^{7}}

\int \frac{1}{u ^{6}} d u = - \frac{1}{5 u ^{5}}

= - \frac{1}{9 u ^{9}} - (- \frac{2}{7 u ^{7}}) - \frac{1}{5 u ^{5}}

Substitute back

u = sin (x)

= - \frac{1}{9 sin ^{9} ( x )} - (- \frac{2}{7 sin ^{7} ( x )}) - \frac{1}{5 sin ^{5} ( x )}

Simplify

- \frac{1}{9 sin ^{9} ( x )} - (- \frac{2}{7 sin ^{7} ( x )}) - \frac{1}{5 sin ^{5} ( x )} : - \frac{1}{9} csc^{9} (x) + \frac{2}{7} csc^{7} (x) - \frac{1}{5} csc^{5} (x)

= - \frac{1}{9} csc^{9} (x) + \frac{2}{7} csc^{7} (x) - \frac{1}{5} csc^{5} (x)

Add a constant to the solution

= - \frac{1}{9} csc^{9} (x) + \frac{2}{7} csc^{7} (x) - \frac{1}{5} csc^{5} (x) + C

What is the integral of csc^5(x)cot^5(x) ?
The integral of csc^5(x)cot^5(x) is -1/9 csc^9(x)+2/7 csc^7(x)-1/5 csc^5(x)+C