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Integrals

Question
CBSEENMA12032616

Show that:
integral subscript 0 superscript straight pi over 2 end superscript fraction numerator straight x space dx over denominator sinx space plus space cosx end fraction space equals space fraction numerator straight pi over denominator 2 square root of 2 end fraction log left parenthesis square root of 2 plus 1 right parenthesis

Solution

Let I = integral subscript 0 superscript straight pi over 2 end superscript fraction numerator xdx over denominator sinx plus cosx end fraction               ...(1)
therefore space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator begin display style straight pi over 2 end style minus straight x over denominator sin open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx space space space space space space open square brackets because space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space 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
          equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator begin display style straight pi over 2 minus straight x end style over denominator cosx plus sinx end fraction dx space equals space straight pi over 2 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator cosx plus sinx end fraction space minus space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator xdx over denominator cosx plus sinx end fraction
therefore space space space straight I space equals space space straight pi over 2 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator cosx plus sinx end fraction space minus 1                           [because space space of space left parenthesis 1 right parenthesis]
therefore space space space space 2 space straight I space equals space straight pi over 2 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator cosx plus sinx end fraction dx                                             ...(2)
Let straight I subscript 1 space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator cosx plus sinx end fraction dx           
Put tan straight x over 2 space equals space straight t space space or space space straight x over 2 space equals space tan to the power of 1 straight t space space or space space straight x space equals space 2 space tan to the power of 1 straight t space space space space rightwards double arrow space space space space space dx space space equals fraction numerator 2 over denominator 1 plus straight t squared end fraction dt
        sinx space equals space fraction numerator 2 space tan begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals fraction numerator 2 space straight t over denominator 1 plus straight t squared end fraction space and space cosx space equals space fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction space
                When x = 0, t = tan 0 = 0
When  straight x space equals space straight pi over 2 comma space space straight t space equals space tan straight pi over 4 space equals space 1
therefore space space space space space space straight I subscript 1 space equals space integral subscript 0 superscript 1 fraction numerator begin display style fraction numerator 1 over denominator 1 plus straight t squared end fraction end style dt over denominator begin display style fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end style plus begin display style fraction numerator 2 space straight t over denominator 1 plus straight t squared end fraction end style end fraction space equals space 2 integral subscript 0 superscript 1 fraction numerator dt over denominator 1 minus straight t squared plus 2 straight t end fraction space equals space 2 integral subscript 0 superscript 1 fraction numerator dt over denominator 2 minus left parenthesis straight t squared minus 2 straight t plus 1 right parenthesis end fraction
             equals space 2 integral subscript 0 superscript 1 fraction numerator dt over denominator left parenthesis square root of 2 right parenthesis squared minus left parenthesis straight t minus 1 right parenthesis squared end fraction space equals 2 cross times fraction numerator 1 over denominator 2 square root of 2 end fraction open square brackets log space open vertical bar fraction numerator square root of 2 plus left parenthesis straight t minus 1 right parenthesis over denominator square root of 2 minus left parenthesis straight t minus 1 right parenthesis end fraction close vertical bar close square brackets subscript 0 superscript 1
equals space fraction numerator 1 over denominator square root of 2 end fraction open square brackets log space fraction numerator square root of 2 over denominator square root of 2 end fraction minus log fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction close square brackets space equals space minus fraction numerator 1 over denominator square root of 2 end fraction log fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction
equals space fraction numerator 1 over denominator square root of 2 end fraction log fraction numerator square root of 2 plus 1 over denominator square root of 2 minus 1 end fraction equals fraction numerator 1 over denominator square root of 2 end fraction log open parentheses fraction numerator square root of 2 plus 1 over denominator square root of 2 minus 1 end fraction cross times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction close parentheses
equals space fraction numerator 1 over denominator square root of 2 end fraction log open square brackets fraction numerator left parenthesis square root of 2 plus 1 right parenthesis squared over denominator 2 minus 1 end fraction close square brackets space equals space fraction numerator 1 over denominator square root of 2 end fraction log left parenthesis square root of 2 plus 1 right parenthesis squared space equals space fraction numerator 2 over denominator square root of 2 end fraction log left parenthesis square root of 2 plus 1 right parenthesis
therefore space space space space space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator cosx plus sinx end fraction space equals space fraction numerator 2 over denominator square root of 2 end fraction log left parenthesis square root of 2 plus 1 right parenthesis
therefore space space space space space space space equals space fraction numerator straight pi over denominator 2 square root of 2 end fraction log left parenthesis square root of 2 plus 1 right parenthesis        

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