Question

Evaluate:
$integral subscript 0 superscript straight pi fraction numerator straight x space over denominator 1 plus sinx end fraction dx space equals space straight pi$

Solution

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

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Integrals

Evaluate:
$integral subscript 0 superscript straight pi fraction numerator straight x space over denominator 1 plus sinx end fraction dx space equals space straight pi$

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