What is the integral of cos(-3x)(sin(-3x))^{-5} ?

The integral of cos(-3x)(sin(-3x))^{-5} is 1/12 csc^4(3x)+C

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integral of cos(-3x)(sin(-3x))^{-5}

\int cos (- 3 x) (sin (- 3 x))^{- 5} d x

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

Solution steps

\int cos (- 3 x) (sin (- 3 x))^{- 5} d x

Apply u-substitution

: \int - \frac{cos ( u )}{3 sin ^{5} ( u )} d u

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

Apply u-substitution

: \int \frac{1}{v ^{5}} d v

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

Apply the Power Rule

: - \frac{1}{4 v ^{4}}

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of cos(-3x)(sin(-3x))^{-5} ?
The integral of cos(-3x)(sin(-3x))^{-5} is 1/12 csc^4(3x)+C