What is the integral of cot(2x)csc^{5/2}(2x) ?

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

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integral of cot(2x)csc^{5/2}(2x)

\int cot (2 x) csc^{\frac{5}{2}} (2 x) d x

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

Solution steps

\int cot (2 x) csc^{\frac{5}{2}} (2 x) d x

Apply u-substitution

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

Integral of a constant:

\int a d x = a x

= (- \frac{1}{5}) u

Substitute back

u = csc^{\frac{5}{2}} (2 x)

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

Simplify

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

Add a constant to the solution

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

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