What is the integral of csc^4(6w) ?

The integral of csc^4(6w) is 1/6 (-1/3 csc^3(6w)cos(6w)-2/3 cot(6w))+C

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integral of csc^4(6w)

\int csc^{4} (6 w) d w

\frac{1}{6} (- \frac{1}{3} csc^{3} (6 w) cos (6 w) - \frac{2}{3} cot (6 w)) + C

Solution steps

\int csc^{4} (6 w) d w

Apply u-substitution

: \int csc^{4} (u) \frac{1}{6} d u

= \int csc^{4} (u) \frac{1}{6} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{6} \cdot \int csc^{4} (u) d u

Apply Integral Reduction

= \frac{1}{6} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} \cdot \int csc^{2} (u) d u)

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{6} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} (- cot (u)))

Substitute back

u = 6 w

= \frac{1}{6} (- \frac{cos ( 6 w ) csc ^{3} ( 6 w )}{3} + \frac{2}{3} (- cot (6 w)))

Simplify

\frac{1}{6} (- \frac{cos ( 6 w ) csc ^{3} ( 6 w )}{3} + \frac{2}{3} (- cot (6 w))) : \frac{1}{6} (- \frac{1}{3} csc^{3} (6 w) cos (6 w) - \frac{2}{3} cot (6 w))

= \frac{1}{6} (- \frac{1}{3} csc^{3} (6 w) cos (6 w) - \frac{2}{3} cot (6 w))

Add a constant to the solution

= \frac{1}{6} (- \frac{1}{3} csc^{3} (6 w) cos (6 w) - \frac{2}{3} cot (6 w)) + C

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What is the integral of csc^4(6w) ?
The integral of csc^4(6w) is 1/6 (-1/3 csc^3(6w)cos(6w)-2/3 cot(6w))+C