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Integrals

Question
CBSEENMA12032562

Show that:
integral subscript 0 superscript straight pi over 2 end superscript space fraction numerator sin to the power of 4 straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx space equals space straight pi over 4

Solution

Let I =integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 4 straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx                         ...(1) 
Then I = integral subscript 0 superscript straight pi divided by 2 end superscript space fraction numerator sin to the power of 4 open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin to the power of 4 open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos to the power of 4 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 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 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator cos to the power of 4 straight x over denominator cos to the power of 4 straight x plus sin to the power of 4 straight x end fraction dx                   ...(2)

Adding (1) and (2), we get.
          2 space straight I space equals integral subscript 0 superscript straight pi divided by 2 end superscript open square brackets fraction numerator sin to the power of 4 space straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction plus fraction numerator cos to the power of 4 straight x over denominator cos to the power of 4 straight x plus sin to the power of 4 straight x end fraction close square brackets space dx
space space space space space equals integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 4 straight x plus cos to the power of 4 straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction space 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 straight pi over 2 minus 0 space equals space straight pi over 2
therefore space space 2 space 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 space space straight I space equals space straight pi over 4

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