Evaluate the following:
$integral subscript 0 superscript straight pi over 4 end superscript square root of 1 minus sin space 2 straight x end root dx$

Solution

Let I = $integral subscript 0 superscript straight pi over 4 end superscript square root of 1 minus sin space 2 straight x end root dx space equals space integral subscript 0 superscript straight pi over 4 end superscript square root of cos squared straight x plus sin squared straight x minus 2 sinx space cosx end root dx$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space square root of left parenthesis cosx minus sinx right parenthesis squared end root space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis cosx minus sinx right parenthesis space dx$
$equals left square bracket sinx plus cosx right square bracket subscript 0 superscript straight pi over 4 end superscript space equals space open parentheses sin straight pi over 4 plus cos straight pi over 4 close parentheses minus left parenthesis sin 0 plus cos 0 right parenthesis equals space open parentheses fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space minus space left parenthesis 0 plus 1 right parenthesis space equals space fraction numerator 2 over denominator square root of 2 end fraction minus 1 space equals square root of 2 minus 1.$

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Integrals

Evaluate the following:
$integral subscript 0 superscript straight pi over 4 end superscript square root of 1 minus sin space 2 straight x end root dx$

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