Question

The value of $square root of 2 space integral fraction numerator sin space straight x space dx over denominator sin space open parentheses straight x minus begin display style straight pi over 4 end style close parentheses end fraction space is$

$space straight x plus space log open vertical bar cos space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus space straight C$
$straight x space minus log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c$
$straight x space plus space log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c$
$straight x space plus space log space open vertical bar cos space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c$

Solution

straight x space plus space log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c

square root of 2 space integral fraction numerator sin space straight x space dx over denominator sin space open parentheses straight x minus begin display style straight pi over 4 end style close parentheses end fraction space equals space square root of 2 integral fraction numerator sin space open parentheses straight x minus begin display style straight pi over 4 end style plus begin display style straight pi over 4 end style close parentheses dx over denominator sin space open parentheses straight x minus space begin display style straight pi over 4 end style close parentheses end fraction space equals space square root of 2 space integral open parentheses cos space straight pi over 4 space plus space cot space open parentheses straight x minus space straight pi over 4 close parentheses space sin space straight pi over 4 close parentheses space dx space equals space integral space dx space plus space integral cot space open parentheses straight x space minus space straight pi over 4 close parentheses dx equals space straight x space plus space In space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c

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