Evaluate the following definite integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of sin space 2 straight x end root end fraction dx.$

Solution

Let
$straight I space equals space integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of sin space 2 straight x end root end fraction dx space equals space integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of 1 minus left parenthesis 1 minus sin space 2 straight x right parenthesis end root end fraction dx$
$equals space integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of 1 minus left parenthesis 1 minus sin space 2 straight x right parenthesis end root end fraction dx space equals space integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of 1 minus left parenthesis sinx minus cosx right parenthesis squared end root end fraction dx$
Put sin x – cos x = y,   ∴ (cos x + sin x) dx = dy
When    $straight x space equals space straight pi over 6 comma space space straight y space equals space sin straight pi over 6 minus cos straight pi over 6 space equals space 1 half minus fraction numerator square root of 3 over denominator 2 end fraction$
When    $straight x space equals space straight pi over 3 comma space space space straight y space equals space sin straight pi over 3 minus cos straight pi over 3 space equals space fraction numerator square root of 3 over denominator 2 end fraction minus 1 half$
$therefore space space space space space space space space space space space space space space space space space space space straight I space space equals space integral subscript fraction numerator 1 over denominator square root of 2 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction end subscript superscript fraction numerator square root of 3 over denominator 2 end fraction minus 1 half end superscript space space fraction numerator 1 over denominator square root of 1 minus straight t squared end root end fraction dt space equals space open square brackets sin to the power of negative 1 end exponent straight t close square brackets subscript 1 half minus fraction numerator square root of 3 over denominator 2 end fraction end subscript superscript fraction numerator square root of 3 over denominator 2 end fraction minus 1 half end superscript$
   $sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction minus 1 half close parentheses space minus space sin to the power of negative 1 end exponent open parentheses 1 half minus fraction numerator square root of 3 over denominator 2 end fraction close parentheses$
$equals space sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction minus 1 half close parentheses space plus space sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction minus 1 half close parentheses space equals space 2 sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 minus 1 over denominator 2 end fraction close parentheses$

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Integrals

Evaluate the following definite integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator sinx plus cosx over denominator square root of sin space 2 straight x end root end fraction dx.$

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