What is the integral of 9csc(x)(cot(x)-csc(x)) ?

The integral of 9csc(x)(cot(x)-csc(x)) is 9(-csc(x)+cot(x))+C

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integral of 9csc(x)(cot(x)-csc(x))

\int 9 csc (x) (cot (x) - csc (x)) d x

9 (- csc (x) + cot (x)) + C

Solution steps

\int 9 csc (x) (cot (x) - csc (x)) d x

Take the constant out:

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

= 9 \cdot \int csc (x) (cot (x) - csc (x)) d x

Expand

csc (x) (cot (x) - csc (x)) : csc (x) cot (x) - csc^{2} (x)

= 9 \cdot \int csc (x) cot (x) - csc^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 9 (\int csc (x) cot (x) d x - \int csc^{2} (x) d x)

\int csc (x) cot (x) d x = - csc (x)

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

= 9 (- csc (x) - (- cot (x)))

Simplify

= 9 (- csc (x) + cot (x))

Add a constant to the solution

= 9 (- csc (x) + cot (x)) + C

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What is the integral of 9csc(x)(cot(x)-csc(x)) ?
The integral of 9csc(x)(cot(x)-csc(x)) is 9(-csc(x)+cot(x))+C