What is the integral of csc^4(9x) ?

The integral of csc^4(9x) is 1/9 (-1/3 csc^3(9x)cos(9x)-2/3 cot(9x))+C

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integral of csc^4(9x)

\int csc^{4} (9 x) d x

\frac{1}{9} (- \frac{1}{3} csc^{3} (9 x) cos (9 x) - \frac{2}{3} cot (9 x)) + C

Solution steps

\int csc^{4} (9 x) d x

Apply u-substitution

= \int csc^{4} (u) \frac{1}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int csc^{4} (u) d u

Apply Integral Reduction

= \frac{1}{9} (- \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}{9} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} (- cot (u)))

Substitute back

u = 9 x

= \frac{1}{9} (- \frac{cos ( 9 x ) csc ^{3} ( 9 x )}{3} + \frac{2}{3} (- cot (9 x)))

Simplify

\frac{1}{9} (- \frac{cos ( 9 x ) csc ^{3} ( 9 x )}{3} + \frac{2}{3} (- cot (9 x))) : \frac{1}{9} (- \frac{1}{3} csc^{3} (9 x) cos (9 x) - \frac{2}{3} cot (9 x))

= \frac{1}{9} (- \frac{1}{3} csc^{3} (9 x) cos (9 x) - \frac{2}{3} cot (9 x))

Add a constant to the solution

= \frac{1}{9} (- \frac{1}{3} csc^{3} (9 x) cos (9 x) - \frac{2}{3} cot (9 x)) + C

What is the integral of csc^4(9x) ?
The integral of csc^4(9x) is 1/9 (-1/3 csc^3(9x)cos(9x)-2/3 cot(9x))+C