What is the integral of (csc^6(2x))/(sqrt(cot(2x))) ?

The integral of (csc^6(2x))/(sqrt(cot(2x))) is -sqrt(cot(2x))-2/5 cot^{5/2}(2x)-1/9 cot^{9/2}(2x)+C

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integral of (csc^6(2x))/(sqrt(cot(2x)))

\int \frac{csc ^{6} ( 2 x )}{cot ( 2 x )} d x

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

Solution steps

\int \frac{csc ^{6} ( 2 x )}{cot ( 2 x )} d x

Apply u-substitution

: \int - 1 - 2 u^{4} - u^{8} d u

= \int - 1 - 2 u^{4} - u^{8} 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 1 d u - \int 2 u^{4} d u - \int u^{8} d u

\int 1 d u = u

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

\int u^{8} d u = \frac{u ^{9}}{9}

= - u - \frac{2 u ^{5}}{5} - \frac{u ^{9}}{9}

Substitute back

u = cot (2 x)

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

Simplify

- cot (2 x) - \frac{2 ( cot ( 2 x ) ) ^{5}}{5} - \frac{( cot ( 2 x ) ) ^{9}}{9} : - cot (2 x) - \frac{2}{5} cot^{\frac{5}{2}} (2 x) - \frac{1}{9} cot^{\frac{9}{2}} (2 x)

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

Add a constant to the solution

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

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What is the integral of (csc^6(2x))/(sqrt(cot(2x))) ?
The integral of (csc^6(2x))/(sqrt(cot(2x))) is -sqrt(cot(2x))-2/5 cot^{5/2}(2x)-1/9 cot^{9/2}(2x)+C