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Integrals

Question
CBSEENMA12032589
Wired Faculty App

Show that integral subscript 0 superscript straight pi over 2 end superscript space log space tanx space dx space equals space 0

Solution

Let I = integral subscript 0 superscript straight pi over 2 end superscript space log space tanx space dx
 therefore space space space space space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript space log space tan 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 dx close square brackets
therefore        straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript log space cotx space dx space equals space integral subscript 0 superscript straight pi divided by 2 end superscript log fraction numerator 1 over denominator tan space straight x end fraction dx space equals space integral subscript 0 superscript straight pi divided by 2 end superscript log space 1 space dx space minus space integral subscript 0 superscript straight pi divided by 2 end superscript log space tanx space dx
  = 0 -  I
therefore space space space space 2 space straight I space equals space 0 space space space space space space space rightwards double arrow space space space space straight I space equals space 0 therefore space space space space space space integral subscript 0 superscript straight pi divided by 2 end superscript log space tanx space dx space equals space 0

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