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Integrals

Question
CBSEENMA12032670

Evaluate integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx.

Solution

Let I = integral subscript 0 superscript straight pi over 2 end superscript space log space sinx space dx                             ...(1)
therefore space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript log space sin open parentheses straight pi over 2 minus straight x close parentheses space dx               open square brackets because space 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
or     straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript space log space cosx 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 left parenthesis log space sinx space plus space log space cosx right parenthesis space dx space equals space integral subscript 0 superscript straight pi over 2 end superscript log space left parenthesis sinx space cosx right parenthesis space dx
            equals space integral subscript 0 superscript straight pi over 2 end superscript log space open parentheses fraction numerator 2 space sinx space cos space straight x over denominator 2 end fraction close parentheses dx space equals space integral subscript 0 superscript straight pi over 2 end superscript log space open parentheses fraction numerator sin space 2 straight x over denominator 2 end fraction close parentheses dx
therefore space space space 2 space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript space log space sin space 2 straight x space dx space space minus space integral subscript 0 superscript straight pi over 2 end superscript space log space 2 space dx
therefore space space space 2 space straight I space equals space straight I subscript 1 minus space straight I subscript 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
where   straight I subscript 2 space equals space integral subscript 0 superscript straight pi over 2 end superscript log space 2 space dx space equals space log space 2 space integral subscript 0 superscript straight pi over 2 end superscript 1. space dx
                 equals space log space 2. space open square brackets straight x close square brackets subscript 0 superscript straight pi over 2 end superscript space equals space space log space 2. space open square brackets straight pi over 2 minus 0 close square brackets space equals space straight pi over 2 space log space 2
and  straight I subscript 1 space equals space integral subscript 0 superscript straight pi over 2 end superscript space log space sin space 2 straight x space dx
Put      2 x = t          or  straight x space equals space straight t over 2 space space space space space space space space space space space space space space space space space rightwards double arrow space space space space dx space equals space 1 half dt
When   x = 0,  t  = 0
When    straight x space equals space straight pi over 2 comma space space space straight t space equals space straight pi
therefore space space space straight I subscript 1 space equals space integral subscript 0 superscript straight pi log space sint. space 1 half dt space equals space 1 half integral subscript 0 superscript straight pi log space sint space dt space equals space 1 half integral subscript 0 superscript straight pi log space sinx space dx
                                                                               open square brackets because space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript straight a superscript straight b straight f left parenthesis straight t right parenthesis space dt close square brackets
           equals space 1 half. space 2 space integral subscript 0 superscript straight pi over 2 end superscript log. space sinx space dx space equals space integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx
                                          open square brackets because space space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space 2 integral subscript 0 superscript straight a over 2 end superscript straight f left parenthesis straight x right parenthesis space dx space when space straight f left parenthesis straight a minus straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis close square brackets
therefore space space space space straight I subscript 1 space equals space 1
therefore space space space space from space left parenthesis 3 right parenthesis comma space space 2 space space straight I space equals space straight I space minus space straight pi over 2 log space 2 space space space rightwards double arrow space space straight I space equals space minus straight pi over 2 space log space 2 space
rightwards double arrow space space space integral subscript 0 superscript straight pi over 2 end superscript space log space sinx space dx space equals space minus straight pi over 2 log space 2
Cor. integral subscript 0 superscript straight pi over 2 end superscript log space cosx space dx space equals space integral subscript 0 superscript straight pi over 2 end superscript log space cos open parentheses straight pi over 2 minus straight x close parentheses dx space equals space integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx space equals space minus straight pi over 2 log space 2

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