Evaluate the indefinite integral.  integrate of cot(x)ln(sin(x))dx

Expert Answers
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You should remember that `cot x = cos x/sin x,`  hence, you may write the integrand such that:

`int (cos x/sin x)ln(sin x)dx`

You should come pu with the following substitution such that:

`sin x = t => cos x dx = dt`

Changing the variable yields:

`int (cos x/sin x)ln(sin x)dx = int (ln t)/t dt`

You should use the next substitution such that:

`ln t = u => 1/t dt = du`

`int (ln t)/t dt = int udu`

`int udu = u^2/2 + c`

Substituting back `ln t`  for u yields:

`int (ln t)/t dt = (ln^2 t)/2 + c`

Substituting back `sin x`  for t yields:

`int (cos x/sin x)ln(sin x)dx = (ln^2 (sin x))/2 + c`

Hence, evaluating the given integral using substitution yields `int (cos x/sin x)ln(sin x)dx = (ln^2 (sin x))/2 + c.`