Question

$integral fraction numerator dx over denominator cos space straight x space minus space sin space straight x end fraction$ is equal to
$fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator begin display style straight pi end style over denominator 8 end fraction close parentheses close vertical bar space plus straight C$
$fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar cot open parentheses straight x over 2 close parentheses close vertical bar space plus straight C$
$fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C$
$fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C$

Solution

fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C

integral fraction numerator dx over denominator cos space straight x space minus space sin space straight x end fraction space equals space fraction numerator 1 over denominator square root of 2 end fraction integral fraction numerator 1 over denominator cos begin display style space end style begin display style open parentheses straight x space plus straight pi over 4 close parentheses end style end fraction space dx equals space fraction numerator 1 over denominator square root of 2 end fraction integral space sec space open parentheses space straight x plus straight pi over 4 space close parentheses dx space equals space fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar space tan space open parentheses straight x over 2 space plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C

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