What is the integral of cot(3s)csc^4(3s) ?

The integral of cot(3s)csc^4(3s) is -1/12 csc^4(3s)+C

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integral of cot(3s)csc^4(3s)

\int cot (3 s) csc^{4} (3 s) d s

- \frac{1}{12} csc^{4} (3 s) + C

Solution steps

\int cot (3 s) csc^{4} (3 s) d s

Apply u-substitution

= \int cot (u) csc^{4} (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 cot (u) csc^{4} (u) d u

Rewrite using trig identities

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

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{1}{v ^{5}} d v

Apply the Power Rule

= \frac{1}{3} (- \frac{1}{4 v ^{4}})

Substitute back

= \frac{1}{3} (- \frac{1}{4 sin ^{4} ( 3 s )})

Simplify

\frac{1}{3} (- \frac{1}{4 sin ^{4} ( 3 s )}) : - \frac{1}{12} csc^{4} (3 s)

= - \frac{1}{12} csc^{4} (3 s)

Add a constant to the solution

= - \frac{1}{12} csc^{4} (3 s) + C

What is the integral of cot(3s)csc^4(3s) ?
The integral of cot(3s)csc^4(3s) is -1/12 csc^4(3s)+C