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Integrals

Question
CBSEENMA12032555

Prove that: integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus tanx end fraction space equals space straight pi over 4.

Solution

Let I = integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus tanx end fraction dx space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus begin display style fraction numerator sinx space over denominator cosx end fraction end style end fraction dx
therefore space space space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cosx over denominator cosx plus sinx end fraction dx                              ...(1)
              equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos space open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator cos open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus sin open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx space space space 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 space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x over denominator sin space straight x space plus space cos space straight x space end fraction dx                               ...(2)
Adding (1) and (2), we get,
                      2 straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator cos space straight x over denominator cos space straight x space plus space sin space straight x end fraction plus fraction numerator sin space straight x over denominator sin space straight x space plus space cos space straight x end fraction close parentheses dx space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cosx plus sinx over denominator cosx plus sinx end fraction dx space equals integral subscript 0 superscript straight pi over 2 end superscript 1 space dx
      equals space open square brackets straight x close square brackets subscript 0 superscript straight pi over 2 end superscript space equals straight pi over 2 minus 0
therefore space space space space space 2 straight I space equals space straight pi over 2 space space space space space space space space space space rightwards double arrow space space space 1 space equals space straight pi over 4

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