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Integrals

Question
CBSEENMA12032567

Show that:
integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus cot space straight x end fraction space equals space straight pi over 4

Solution

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

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