Question

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 2 space log space sinx space minus space log space sin space 2 straight x right parenthesis space dx$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 2 space log space sinx space minus space log space sin space 2 straight x right parenthesis space dx$ ...(1)
$therefore space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript open square brackets 2 space log space sin open parentheses straight pi over 2 minus straight x close parentheses minus log space sin 2 open parentheses straight pi over 2 minus straight x close parentheses close square brackets dx$    $open square brackets because space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript space left parenthesis 2 space log space cos space straight x space minus space log space sin space 2 straight x right parenthesis space dx$ ...(2)
Adding (1) and (2), we get,
   $2 space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript open square brackets 2 left parenthesis log space sinx space plus space log space cosx right parenthesis space minus space 2 space log space sin space 2 straight x close square brackets dx$
   $equals space integral subscript 0 superscript straight pi over 2 end superscript open square brackets 2 space log space open parentheses fraction numerator 2 space sinx space cosx over denominator 2 end fraction close parentheses minus 2 space log space sin 2 straight x close square brackets dx space equals space integral subscript 0 superscript straight pi over 2 end superscript open square brackets 2 space log space fraction numerator sin 2 straight x over denominator 2 end fraction minus 2 space log space sin 2 straight x close square brackets dx equals space integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 2 space log space sin 2 straight x minus space 2 space log space 2 space minus 2 space log space sin space 2 straight x right square bracket space dx equals space minus space 2 space log space 2 space integral subscript 0 superscript straight pi over 2 end superscript 1 space dx space equals space minus 2 space log space 2 open square brackets straight x close square brackets subscript 0 superscript straight pi over 2 end superscript space equals space minus 2 space log space 2 open square brackets straight pi over 2 minus 0 close square brackets space equals space minus straight pi space log space 2 therefore space 2 space straight I space equals space minus straight pi space space log space 2 space rightwards double arrow space space space straight I space equals space minus straight pi over 2 log space 2$

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Integrals

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 2 space log space sinx space minus space log space sin space 2 straight x right parenthesis space dx$

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