What is the integral of csc^2(x)cot(x)ln((csc(x))/2) ?

The integral of csc^2(x)cot(x)ln((csc(x))/2) is -1/2 csc^2(x)ln(1/2 csc(x))+1/4 csc^2(x)+C

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integral of csc^2(x)cot(x)ln((csc(x))/2)

\int csc^{2} (x) cot (x) ln (\frac{csc ( x )}{2}) d x

- \frac{1}{2} csc^{2} (x) ln (\frac{1}{2} csc (x)) + \frac{1}{4} csc^{2} (x) + C

Solution steps

\int csc^{2} (x) cot (x) ln (\frac{csc ( x )}{2}) d x

Apply Integration By Parts

: - \frac{1}{2} csc^{2} (x) ln (\frac{1}{2} csc (x)) - \int \frac{1}{2} csc^{2} (x) cot (x) d x

= - \frac{1}{2} csc^{2} (x) ln (\frac{1}{2} csc (x)) - \int \frac{1}{2} csc^{2} (x) cot (x) d x

\int \frac{1}{2} csc^{2} (x) cot (x) d x = - \frac{1}{4} csc^{2} (x)

= - \frac{1}{2} csc^{2} (x) ln (\frac{1}{2} csc (x)) - (- \frac{1}{4} csc^{2} (x))

Simplify

= - \frac{1}{2} csc^{2} (x) ln (\frac{1}{2} csc (x)) + \frac{1}{4} csc^{2} (x)

Add a constant to the solution

= - \frac{1}{2} csc^{2} (x) ln (\frac{1}{2} csc (x)) + \frac{1}{4} csc^{2} (x) + C

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What is the integral of csc^2(x)cot(x)ln((csc(x))/2) ?
The integral of csc^2(x)cot(x)ln((csc(x))/2) is -1/2 csc^2(x)ln(1/2 csc(x))+1/4 csc^2(x)+C