Question

Evaluate: $integral subscript 0 superscript straight a space sin to the power of negative 1 end exponent space open parentheses square root of fraction numerator straight x over denominator straight a plus straight x end fraction end root close parentheses dx.$

Solution

Let
$straight I space equals space integral subscript 0 superscript straight a space sin to the power of negative 1 end exponent open parentheses square root of fraction numerator straight x over denominator straight a plus straight x end fraction end root close parentheses dx$
Put $square root of fraction numerator straight x over denominator straight a plus straight x end fraction end root space equals space sin space straight theta space space space or space space fraction numerator straight x over denominator straight a plus straight x end fraction space equals space sin squared straight theta$
$therefore space space space space straight x space space equals straight a space sin squared space straight theta space plus space straight x space sin squared space straight theta space space space space space space rightwards double arrow space space space space space straight x left parenthesis 1 minus sin squared straight theta right parenthesis space equals space straight a space sin squared straight theta$
$rightwards double arrow space space space space straight x space cos squared straight theta space equals space straight a space sin squared straight theta space space space space space space space rightwards double arrow space space space straight x space equals space straight a space tan squared straight theta$
$therefore space space space space space space space space space dx space equals space 2 space straight a space tanθ space space sec squared straight theta space dθ$
When x = 0, $sin space straight theta space equals space 0$    $rightwards double arrow space space space straight theta space equals space 0$
When x = a, $sin space straight theta space equals space square root of fraction numerator straight a over denominator straight a plus straight a end fraction end root space equals space fraction numerator 1 over denominator square root of 2 end fraction space space space space rightwards double arrow space space space space straight theta space equals space straight pi over 4$
$therefore$ $straight I space equals space integral subscript 0 superscript straight pi over 4 end superscript space sin to the power of negative 1 end exponent left parenthesis sin space straight theta right parenthesis. space 2 space straight a space tanθ space sec squared straight theta space dθ space equals space 2 straight a integral subscript 0 superscript straight pi over 4 end superscript space straight theta. space left parenthesis tanθ space sec squared straight theta right parenthesis space dθ$
   $equals space 2 space straight a open curly brackets open square brackets straight theta. space fraction numerator tan squared straight theta over denominator 2 end fraction close square brackets subscript 0 superscript straight pi over 4 end superscript space minus space integral subscript 0 superscript straight pi over 4 end superscript 1. space fraction numerator tan squared space straight theta over denominator 2 end fraction dθ close curly brackets$
   $equals space straight a space open square brackets straight theta space tan squared straight theta close square brackets subscript 0 superscript straight pi over 4 end superscript space minus space straight a integral subscript 0 superscript straight pi over 4 end superscript space left parenthesis sec squared straight theta minus 1 right parenthesis space dθ equals space straight a open square brackets straight theta space tan squared straight theta close square brackets subscript 0 superscript straight pi over 4 end superscript space minus space straight a space open square brackets tanθ space minus space straight theta close square brackets subscript 0 superscript straight pi over 4 end superscript equals space straight a open square brackets straight pi over 4 cross times tan squared straight pi over 4 minus 0 close square brackets space minus space straight a open square brackets open parentheses tan straight pi over 4 minus straight pi over 4 close parentheses minus left parenthesis tan space 0 space minus 0 right parenthesis close square brackets equals space straight a cross times straight pi over 4 cross times 1 minus straight a open parentheses 1 minus straight pi over 4 close parentheses space equals space πa over 4 minus straight a plus πa over 4 space equals space fraction numerator straight pi space straight a over denominator 2 end fraction minus straight a space equals space straight a open parentheses straight pi over 2 minus 1 close parentheses space equals space fraction numerator straight a left parenthesis straight pi minus 2 right parenthesis over denominator 2 end fraction$

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Integrals

Evaluate: $integral subscript 0 superscript straight a space sin to the power of negative 1 end exponent space open parentheses square root of fraction numerator straight x over denominator straight a plus straight x end fraction end root close parentheses dx.$

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