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

The integral of (-csc(x))/(csc(x)-cot(x)) is cot(x/2)+C

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

\int \frac{- csc ( x )}{csc ( x ) - cot ( x )} d x

cot (\frac{x}{2}) + C

Solution steps

\int \frac{- csc ( x )}{csc ( x ) - cot ( x )} d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{u ^{2}} d u

= - \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= - (- \frac{1}{u})

Substitute back

u = tan (\frac{x}{2})

= - (- \frac{1}{tan ( \frac{x}{2} )})

Simplify

- (- \frac{1}{tan ( \frac{x}{2} )}) : cot (\frac{x}{2})

= cot (\frac{x}{2})

Add a constant to the solution

= cot (\frac{x}{2}) + C

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What is the integral of (-csc(x))/(csc(x)-cot(x)) ?
The integral of (-csc(x))/(csc(x)-cot(x)) is cot(x/2)+C