Question

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x over denominator sin space straight x plus space cos space straight x end fraction dx space equals space straight pi over 4$

Solution

Let I = $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin space straight x over denominator sin space straight x space plus space cos space straight x end fraction dx$ ...(1)
$therefore space space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin space open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos space open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction space space space space space space space open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx space space space close square brackets$
$therefore space space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator cosx over denominator cosx plus sinx end fraction dx$ ...(2)
Adding (1) and (2), we get
$2 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript open square brackets fraction numerator sin space straight x over denominator sin space straight x plus space cos space straight x end fraction plus fraction numerator cosx over denominator cosx plus sinx end fraction close square brackets dx$
$equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sinx plus cosx over denominator sinx plus cosx end fraction dx space equals integral subscript 0 superscript straight pi divided by 2 end superscript 1 space dx space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript space equals space straight pi over 2 minus 0 space equals space straight pi over 2$
$therefore space space 2 space space straight I space equals space space straight pi over 2 space space space space space rightwards double arrow space space space space space space straight I space equals space straight pi over 4$

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Integrals

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x over denominator sin space straight x plus space cos space straight x end fraction dx space equals space straight pi over 4$

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