Question

By using the properties of definite integrals, evaluate the following:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator square root of cosx over denominator square root of cosx plus square root of sinx end fraction dx$

Solution

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

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Integrals

By using the properties of definite integrals, evaluate the following:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator square root of cosx over denominator square root of cosx plus square root of sinx end fraction dx$

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