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Asked by subhbksc1 27th August 2020, 2:50 PM
Answered by Expert
Answer:
Integral 0 to pi {xsinx/1+cos^2x}
L e t space I equals integral subscript 0 superscript straight pi fraction numerator x sin x over denominator 1 plus cos squared x end fraction d x
rightwards double arrow I equals integral subscript 0 superscript straight pi fraction numerator left parenthesis x minus straight pi right parenthesis sin x over denominator 1 plus cos squared x end fraction d x space space space.... space S i n c e space integral subscript a superscript b f left parenthesis x right parenthesis d x equals integral subscript a superscript b f left parenthesis a plus b minus x right parenthesis d x
rightwards double arrow 2 I equals integral subscript 0 superscript straight pi fraction numerator negative πsin x over denominator 1 plus cos squared x end fraction d x
L e t space cos x equals t rightwards double arrow negative sin x d x equals d t
W h e n space x equals 0 rightwards double arrow t equals 1
W h e n space x equals straight pi rightwards double arrow straight t equals negative 1
rightwards double arrow 2 I equals integral subscript negative 1 end subscript superscript 1 fraction numerator straight pi d t over denominator 1 plus t squared end fraction equals 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 equals straight pi open square brackets straight pi over 4 plus straight pi over 4 close square brackets equals straight pi squared over 2
rightwards double arrow straight I equals straight pi squared over 4
Answered by Expert 28th August 2020, 10:52 AM
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