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

The integral of cot(x)ln(csc(x)) is ln(csc(x))ln|sin(x)|+1/2 ln|sin(x)|^2+C

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integral of cot(x)ln(csc(x))

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

ln (csc (x)) ln ∣ sin (x) ∣ + \frac{1}{2} ln ∣ sin (x) ∣^{2} + C

Solution steps

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

Apply Integration By Parts

: ln (csc (x)) ln ∣ sin (x) ∣ - \int - cot (x) ln ∣ sin (x) ∣ d x

= ln (csc (x)) ln ∣ sin (x) ∣ - \int - cot (x) ln ∣ sin (x) ∣ d x

\int - cot (x) ln ∣ sin (x) ∣ d x = - \frac{ln ∣ sin ( x ) ∣ ^{2}}{2}

= ln (csc (x)) ln ∣ sin (x) ∣ - (- \frac{ln ∣ sin ( x ) ∣ ^{2}}{2})

Simplify

ln (csc (x)) ln ∣ sin (x) ∣ - (- \frac{ln ∣ sin ( x ) ∣ ^{2}}{2}) : ln (csc (x)) ln ∣ sin (x) ∣ + \frac{1}{2} ln ∣ sin (x) ∣^{2}

= ln (csc (x)) ln ∣ sin (x) ∣ + \frac{1}{2} ln ∣ sin (x) ∣^{2}

Add a constant to the solution

= ln (csc (x)) ln ∣ sin (x) ∣ + \frac{1}{2} ln ∣ sin (x) ∣^{2} + C

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What is the integral of cot(x)ln(csc(x)) ?
The integral of cot(x)ln(csc(x)) is ln(csc(x))ln|sin(x)|+1/2 ln|sin(x)|^2+C