Evaluate the following definite integral
$integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cosec space straight x space dx$

Solution

Let I = $integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cos e c space x space d x space equals subscript blank superscript blank$ $open square brackets table row log cell open vertical bar cosecx minus cotx close vertical bar end cell end table close square brackets subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript$
$equals space log space open vertical bar cose space straight pi over 4 minus cot straight pi over 4 close vertical bar space minus space log space open vertical bar cosec straight pi over 6 minus cot straight pi over 6 close vertical bar equals space log space open vertical bar square root of 2 minus 1 close vertical bar space minus space log space open vertical bar 2 minus square root of 3 close vertical bar equals space log space left parenthesis square root of 2 minus 1 right parenthesis space minus space log space left parenthesis 2 minus square root of 3 right parenthesis space equals space log open parentheses fraction numerator square root of 2 minus 1 over denominator 2 minus square root of 3 end fraction close parentheses$

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Integrals

Evaluate the following definite integral
$integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cosec space straight x space dx$

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