Question

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin to the power of begin display style 3 over 2 end style end exponent straight x over denominator sin to the power of begin display style 3 over 2 end style end exponent straight x plus cos to the power of begin display style 3 over 2 end style end exponent straight x end fraction dx space$

Solution

Let I = $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 3 divided by 2 end exponent straight x over denominator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x end fraction dx$ ...(1)
Then $straight I space equals integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 3 divided by 2 end exponent open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin to the power of 3 divided by 2 end exponent open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos to the power of 3 divided by 2 end exponent open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx$ $open square brackets because space space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space space equals integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator cos to the power of 3 divided by 2 end exponent straight x over denominator cos to the power of 3 divided by 2 end exponent straight x plus sin to the power of 3 divided by 2 end exponent straight x end fraction dx.$ ...(2)
Adding (1) and (2), we get,
$2 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript open square brackets fraction numerator sin to the power of 3 divided by 2 end exponent straight x over denominator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x end fraction plus fraction numerator cos to the power of 3 divided by 2 end exponent straight x over denominator cos to the power of 3 divided by 2 end exponent straight x plus sin to the power of 3 divided by 2 end exponent straight x end fraction close square brackets dx$
$equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x over denominator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x end fraction dx space equals space integral subscript 0 superscript straight pi divided by 2 end superscript 1 space dx space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript space equals space straight pi over 2 minus 0 space equals space straight pi over 2$
$therefore space space space 2 space straight I space equals space straight pi over 2 space space space space space space space rightwards double arrow space space space space space space space straight I space equals space straight pi over 4$

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