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Integrals

Question
CBSEENMA12032641

By using the properties of definite integrals, evaluate the following:
integral subscript 0 superscript straight pi log space left parenthesis 1 plus cosx right parenthesis space dx

Solution

Let I = integral subscript 0 superscript straight pi log left parenthesis 1 plus cosx right parenthesis space dx                                                   ...(1)
Then I = integral subscript 0 superscript straight pi log left parenthesis 1 plus cos stack straight pi minus straight x with bar on top right parenthesis space dx
therefore space space space space straight I space equals space integral subscript 0 superscript straight pi log left parenthesis 1 minus cosx right parenthesis space dx                                              ...(2)
Adding (1) and (2), we get, 
2 space straight I space equals space integral subscript 0 superscript straight pi open square brackets log left parenthesis 1 plus cosx right parenthesis plus log left parenthesis 1 minus cosx right parenthesis close square brackets dx space equals space integral subscript 0 superscript straight pi log open square brackets left parenthesis 1 plus cosx right parenthesis space left parenthesis 1 minus cosx right parenthesis close square brackets dx
       equals space integral subscript 0 superscript straight pi log left parenthesis 1 minus cos squared straight x right parenthesis space dx space equals space integral subscript 0 superscript straight pi log space left parenthesis sin squared straight x right parenthesis space dx space equals space 2 integral subscript 0 superscript straight pi log space sinx space dx
       equals space 2.2 space integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx                   open square brackets because space space log space sin left parenthesis straight pi minus straight x right parenthesis space equals space log space sinx close square brackets
         equals space 4 open parentheses negative straight pi over 2 log 2 close parentheses space equals space minus 2 space straight pi space log space 2
therefore space space straight I space equals space minus straight pi space log space 2

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