Question

$Evaluate colon integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx$

Solution

We need to evaluate the integral
$straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx... left parenthesis 1 right parenthesis Using space the space property space integral straight f left parenthesis straight a minus straight x right parenthesis dx equals integral straight f left parenthesis straight x right parenthesis dx comma space we space have straight I equals space integral subscript 0 superscript straight pi fraction numerator 4 left parenthesis straight pi minus straight x right parenthesis space sin left parenthesis straight pi minus straight x right parenthesis over denominator 1 plus cos squared left parenthesis straight pi minus straight x right parenthesis end fraction dx rightwards double arrow space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight pi space sinx over denominator 1 plus cos squared straight x end fraction dx minus integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx... left parenthesis 2 right parenthesis Adding space equations space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space have comma$
$rightwards double arrow 2 straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx space plus space integral subscript 0 superscript straight pi fraction numerator 4 πsinx over denominator 1 plus cos squared straight x end fraction dx minus integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx rightwards double arrow 2 straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight pi space sinx over denominator 1 plus cos squared straight x end fraction dx rightwards double arrow 2 straight I space equals space 4 straight pi integral subscript 0 superscript straight pi fraction numerator sinx over denominator 1 plus cos squared straight x end fraction dx$
$Substitute space straight t space equals space cosx semicolon space dt space equals space minus sinxdx when space straight x space equals space 0 comma space space straight t space equals 1 when space straight x space equals space straight pi comma space straight t equals negative 1 rightwards double arrow 2 straight I space equals space 4 straight pi integral subscript 1 superscript negative 1 end superscript fraction numerator left parenthesis negative 1 right parenthesis dt over denominator 1 plus straight t squared end fraction rightwards double arrow space straight I space equals space 2 straight pi integral subscript negative 1 end subscript superscript 1 space fraction numerator dt over denominator 1 plus straight t squared end fraction rightwards double arrow straight I space equals 2 cross times 2 straight pi integral subscript 0 superscript 1 fraction numerator dt over denominator 1 plus straight t squared end fraction rightwards double arrow straight I space equals space 2 cross times 2 straight pi open parentheses tan to the power of negative 1 end exponent straight t close parentheses subscript 0 superscript 1 rightwards double arrow space straight I equals space 4 straight pi space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis rightwards double arrow space straight I equals 4 straight pi cross times straight pi over 4 equals straight pi squared$

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Integrals

$Evaluate colon integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx$

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