Question

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sinx space cosx end root end fraction dx.$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx plus cosx over denominator square root of sin space straight x space cosx end root end fraction dx space equals space 4 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx plus cosx over denominator square root of 2 space sinx space cosx end root end fraction dx$
$equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sin space 2 straight x end root end fraction dx space equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx space plus space cosx over denominator square root of 1 minus left parenthesis 1 minus sin space 2 straight x right parenthesis end root end fraction dx$
$equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of 1 minus left parenthesis sin squared straight x plus cos squared straight x minus 2 sinx space cosx right parenthesis end root end fraction dx equals space 4 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x space over denominator square root of 1 minus left parenthesis sinx space space minus cosx right parenthesis squared end root end fraction dx$
Put sin x - cos x = y, ∴ (cos x + sin x) dx = dy When x = 0, y = sin 0 – cos 0 = 0 – 1 = – 1
When $straight x equals space straight pi over 2 comma space space space straight y space equals space sin straight pi over 2 minus cos straight pi over 2 space equals space 1 minus 0 space equals space 1$
$therefore$ $straight I space equals space 4 integral subscript negative 1 end subscript superscript 1 fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction dy space equals space 4 open square brackets sin to the power of negative 1 end exponent space straight y close square brackets subscript negative 1 end subscript superscript 1 space equals space 4 open square brackets sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis space minus space sin to the power of negative 1 end exponent left parenthesis negative 1 right parenthesis close square brackets$
$equals space 4 space open square brackets straight pi over 2 minus open parentheses negative straight pi over 2 close parentheses close square brackets space equals space 4 open parentheses straight pi over 2 plus straight pi over 2 close parentheses space equals space 4 straight pi$

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Integrals

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sinx space cosx end root end fraction dx.$

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