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

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

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integral of (-csc(θ))/(csc(θ)-cot(θ))

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

cot (\frac{θ}{2}) + C

Solution steps

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

Take the constant out:

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

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

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{θ}{2})

= - (- \frac{1}{tan ( \frac{θ}{2} )})

Simplify

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

= cot (\frac{θ}{2})

Add a constant to the solution

= cot (\frac{θ}{2}) + C

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