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

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

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

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

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

Solution steps

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

Expand

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

= \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

= \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)

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

Simplify

= - csc (x) + cot (x)

Add a constant to the solution

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

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