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Integrals

Question
CBSEENMA12032561

Show that:
integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin cubed straight x over denominator sin cubed straight x plus cos cubed 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 cubed straight x over denominator sin cubed straight x plus cos cubed straight x end fraction dx                          ...(1)
Then I = integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin cubed open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin cubed open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos cubed 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 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript space fraction numerator cos cubed straight x over denominator cos cubed straight x plus sin cubed straight x 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 open square brackets fraction numerator sin cubed straight x over denominator sin cubed straight x plus cos cubed straight x end fraction plus fraction numerator cos cubed straight x over denominator cos cubed straight x plus sin cubed straight x end fraction close square brackets dx
space space space space space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin cubed straight x plus cos cubed straight x over denominator sin cubed straight x plus cos cubed straight x end fraction dx space equals space integral subscript 0 superscript straight pi divided by 2 end superscript space 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 space 2 space straight I space equals space straight pi over 2 space space space space rightwards double arrow space space space space space space space straight I space equals space straight pi over 4

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