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Integrals

Question
CBSEENMA12032553

By using the properties of definite integrals, evaluate the following:
integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos to the power of 5 straight x over denominator sin to the power of 5 straight x plus cos to the power of 5 straight x end fraction dx

Solution

Let I = integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator cos to the power of 5 straight x over denominator sin to the power of 5 straight x plus cos to the power of 5 straight x end fraction dx                        ...(1)
Then straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos to the power of 5 open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin to the power of 5 open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos to the power of 5 open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx space space 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 space dx close square brackets
therefore space space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin to the power of 5 straight x over denominator cos to the power of 5 straight x plus sin to the power of 5 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 over 2 end superscript open parentheses fraction numerator cos to the power of 5 straight x over denominator sin to the power of 5 straight x plus cos to the power of 5 straight x end fraction plus fraction numerator sin to the power of 5 straight x over denominator sin to the power of 5 straight x plus cos to the power of 5 straight x end fraction close parentheses dx space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin to the power of 5 straight x plus cos to the power of 5 straight x over denominator sin to the power of 5 straight x plus cos to the power of 5 straight x end fraction dx space equals space 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 divided by 2 end superscript space equals space straight pi over 2 minus 0
 2 space straight I space equals space straight pi over 2 space space space space rightwards double arrow space space space space straight I space equals space straight pi over 4

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