What is the integral of csc^5(3x) ?

The integral of csc^5(3x) is -1/12 csc^4(3x)cos(3x)+1/8 (-cot(3x)csc(3x)+ln|tan((3x)/2)|)+C

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integral of csc^5(3x)

\int csc^{5} (3 x) d x

- \frac{1}{12} csc^{4} (3 x) cos (3 x) + \frac{1}{8} (- cot (3 x) csc (3 x) + ln tan (\frac{3 x}{2})) + C

Solution steps

\int csc^{5} (3 x) d x

Apply u-substitution

: \int csc^{5} (u) \frac{1}{3} d u

= \int csc^{5} (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int csc^{5} (u) d u

Apply Integral Reduction

= \frac{1}{3} (- \frac{cos ( u ) csc ^{4} ( u )}{4} + \frac{3}{4} \cdot \int csc^{3} (u) d u)

\int csc^{3} (u) d u = - \frac{1}{2} cot (u) csc (u) + \frac{1}{2} ln tan (\frac{u}{2})

= \frac{1}{3} (- \frac{cos ( u ) csc ^{4} ( u )}{4} + \frac{3}{4} (- \frac{1}{2} cot (u) csc (u) + \frac{1}{2} ln tan (\frac{u}{2})))

Substitute back

u = 3 x

= \frac{1}{3} (- \frac{cos ( 3 x ) csc ^{4} ( 3 x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (3 x) csc (3 x) + \frac{1}{2} ln tan (\frac{3 x}{2})))

Simplify

\frac{1}{3} (- \frac{cos ( 3 x ) csc ^{4} ( 3 x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (3 x) csc (3 x) + \frac{1}{2} ln tan (\frac{3 x}{2}))) : - \frac{1}{12} csc^{4} (3 x) cos (3 x) + \frac{1}{8} (- cot (3 x) csc (3 x) + ln tan (\frac{3 x}{2}))

= - \frac{1}{12} csc^{4} (3 x) cos (3 x) + \frac{1}{8} (- cot (3 x) csc (3 x) + ln tan (\frac{3 x}{2}))

Add a constant to the solution

= - \frac{1}{12} csc^{4} (3 x) cos (3 x) + \frac{1}{8} (- cot (3 x) csc (3 x) + ln tan (\frac{3 x}{2})) + C

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What is the integral of csc^5(3x) ?
The integral of csc^5(3x) is -1/12 csc^4(3x)cos(3x)+1/8 (-cot(3x)csc(3x)+ln|tan((3x)/2)|)+C