Plz answer the very last sum(after OR)

Asked by lovemaan5500 | 7th Feb, 2020, 09:50: AM

Expert Answer:

straight I equals integral subscript negative straight pi end subscript superscript straight pi fraction numerator 2 straight x open parentheses 1 plus sin space straight x close parentheses over denominator 1 plus cos squared straight x end fraction dx
equals integral subscript negative straight pi end subscript superscript straight pi fraction numerator 2 straight x over denominator 1 plus cos squared straight x end fraction dx plus integral subscript negative straight pi end subscript superscript straight pi fraction numerator 2 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx
straight I subscript 1 equals integral subscript negative straight pi end subscript superscript straight pi fraction numerator 2 straight x over denominator 1 plus cos squared straight x end fraction dx
here space straight f left parenthesis negative straight x right parenthesis space equals space minus straight f left parenthesis straight x right parenthesis
hence space it space is space 0
straight I equals 0 plus integral subscript negative straight pi end subscript superscript straight pi fraction numerator 2 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx
here space space straight f left parenthesis negative straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis
so space re space writing
straight I equals 2 integral subscript 0 superscript straight pi fraction numerator 2 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx... left parenthesis 1 right parenthesis
straight I equals 2 integral subscript 0 superscript straight pi fraction numerator 2 open parentheses straight pi minus straight x close parentheses space sin open parentheses straight pi minus straight x close parentheses over denominator 1 plus cos squared open parentheses straight pi minus straight x close parentheses end fraction dx
straight I equals 2 integral subscript 0 superscript straight pi fraction numerator 2 open parentheses straight pi minus straight x close parentheses space sinx over denominator 1 plus cos squared straight x end fraction dx... left parenthesis 2 right parenthesis
left parenthesis 1 right parenthesis plus left parenthesis 2 right parenthesis
2 straight I equals 2 integral subscript 0 superscript straight pi fraction numerator 2 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx plus 2 integral subscript 0 superscript straight pi fraction numerator 2 open parentheses straight pi minus straight x close parentheses space sinx over denominator 1 plus cos squared straight x end fraction dx
straight I equals integral subscript 0 superscript straight pi fraction numerator 2 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx plus 2 integral subscript 0 superscript straight pi fraction numerator straight pi space sinx over denominator 1 plus cos squared straight x end fraction dx minus 2 integral subscript 0 superscript straight pi fraction numerator straight x space sinx over denominator 1 plus cos squared straight x end fraction dx
straight I equals 2 integral subscript 0 superscript straight pi fraction numerator straight pi space sinx over denominator 1 plus cos squared straight x end fraction dx
put space cosx space equals straight t
straight I equals negative 2 straight pi integral subscript negative 1 end subscript superscript 1 open parentheses fraction numerator 1 over denominator 1 plus straight t squared end fraction close parentheses dx
straight I equals negative 2 straight pi open square brackets tan to the power of negative 1 end exponent straight t close square brackets subscript negative 1 end subscript superscript 1
straight I equals straight pi squared

Answered by Arun | 8th Feb, 2020, 08:53: AM