Question

Show that:
$integral subscript 0 superscript straight a fraction numerator square root of straight x over denominator square root of straight x plus square root of straight a minus straight x end root end fraction space equals space straight a over 2$

Solution

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

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Integrals

Show that:
$integral subscript 0 superscript straight a fraction numerator square root of straight x over denominator square root of straight x plus square root of straight a minus straight x end root end fraction space equals space straight a over 2$

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