Question

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx plus tanx end fraction dx space equals space straight pi open parentheses straight pi over 2 minus 1 close parentheses$

Solution

Let I = $integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx plus tanx end fraction dx$ ...(1)
$therefore space space space space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator left parenthesis straight pi minus straight x right parenthesis space tan left parenthesis straight pi minus straight x right parenthesis over denominator sec left parenthesis straight pi minus straight x right parenthesis plus tan left parenthesis straight pi minus straight x right parenthesis end fraction dx space space space space open square brackets because space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore space space space straight I space equals space integral subscript 0 superscript straight pi open square brackets fraction numerator left parenthesis straight pi minus straight x right parenthesis space left parenthesis negative tanx right parenthesis over denominator negative secx minus tanx end fraction close square brackets dx space space space rightwards double arrow space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator left parenthesis straight pi minus straight x right parenthesis space tanx over denominator secx plus tanx end fraction dx$
$rightwards double arrow space space space space space straight I space equals straight pi integral subscript 0 superscript straight pi fraction numerator tanx over denominator secx plus tanx end fraction dx space minus space integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx plus tanx end fraction dx$
$rightwards double arrow space space straight I space equals space straight pi integral subscript 0 superscript straight pi over 2 end superscript fraction numerator tan space straight x over denominator secx plus tanx end fraction cross times fraction numerator secx minus tanx over denominator secx minus tanx end fraction dx space minus space 1$ [ $because space of space left parenthesis 1 right parenthesis$ ]
$rightwards double arrow space space space space 2 space straight I space equals space straight pi space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator tanx space secx minus tan squared straight x over denominator sec squared straight x minus tan squared straight x end fraction dx space space space rightwards double arrow space space space 2 space straight I space space equals space straight pi integral subscript 0 superscript straight pi fraction numerator secx space tanx minus left parenthesis sec squared straight x minus 1 right parenthesis over denominator 1 end fraction dx rightwards double arrow space space space 2 straight I space equals space straight pi integral subscript 0 superscript straight pi left parenthesis secx space tanx space minus sec squared straight x plus 1 right parenthesis space dx space equals space straight pi left square bracket secx minus tanx plus straight x right square bracket subscript 0 superscript straight pi$
$equals space straight pi open square brackets secπ minus tanπ plus straight pi right parenthesis space minus space left parenthesis sec space 0 space minus tan space 0 space plus 0 right parenthesis close square brackets equals straight pi open square brackets left parenthesis negative 1 minus 0 plus straight pi right parenthesis space minus space left parenthesis 1 minus 0 plus 0 right parenthesis close square brackets$
$rightwards double arrow space space 2 space straight I space equals space straight pi left parenthesis straight pi minus 2 right parenthesis rightwards double arrow space space space space space straight I space equals space straight pi open parentheses straight pi over 2 minus 1 close parentheses$

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Integrals

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx plus tanx end fraction dx space equals space straight pi open parentheses straight pi over 2 minus 1 close parentheses$

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