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INTEGRATE in 0 to pi ------- (tanx)/(tan x + sec x)....

Asked by Dhruv 20th March 2017, 6:23 PM
Answered by Expert
Answer:

begin mathsize 16px style L e t space I equals integral subscript 0 superscript straight pi fraction numerator tanx over denominator tanx plus secx end fraction dx
space space space space space space space equals integral subscript 0 superscript straight pi fraction numerator sinx over denominator 1 plus sinx end fraction dx space space space.... left parenthesis straight i right parenthesis
space space space space space space space equals integral subscript 0 superscript straight pi fraction numerator sin left parenthesis straight pi minus straight x right parenthesis over denominator 1 plus sin left parenthesis straight pi minus straight x right parenthesis end fraction dx
space space space space space space space equals integral subscript 0 superscript straight pi fraction numerator sinx over denominator 1 plus straight x end fraction dx space space space space space space space space.... left parenthesis ii right parenthesis
Adding space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
2 straight I equals integral subscript 0 superscript straight pi fraction numerator sinx over denominator 1 plus sinx end fraction dx space
Using space integral subscript 0 superscript 2 straight a end superscript space straight f left parenthesis straight x right parenthesis space dx equals space integral subscript 0 superscript straight a space left curly bracket straight f left parenthesis straight x right parenthesis plus straight f left parenthesis 2 straight a minus straight x right parenthesis right curly bracket space dx
rightwards double arrow 2 straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript open curly brackets fraction numerator sinx over denominator 1 plus sinx end fraction plus fraction numerator sin left parenthesis straight pi minus straight x right parenthesis over denominator 1 plus sin left parenthesis straight pi minus straight x right parenthesis end fraction close curly brackets dx
rightwards double arrow 2 straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript open curly brackets fraction numerator sinx over denominator 1 plus sinx end fraction plus fraction numerator sinx over denominator 1 plus sinx end fraction close curly brackets dx
rightwards double arrow 2 straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript open curly brackets fraction numerator 2 sinx over denominator 1 plus sinx end fraction close curly brackets dx
rightwards double arrow straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript open curly brackets fraction numerator sinx over denominator 1 plus sinx end fraction close curly brackets dx
rightwards double arrow straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript open curly brackets fraction numerator sinx left parenthesis 1 minus sinx right parenthesis over denominator 1 minus sin squared straight x end fraction close curly brackets dx
rightwards double arrow straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript open curly brackets fraction numerator sinx minus sin squared straight x over denominator cos squared straight x end fraction close curly brackets dx
rightwards double arrow straight I equals integral subscript 0 superscript straight pi divided by 2 end superscript left parenthesis tanx space secx minus tan squared straight x right parenthesis dx
Use space sec squared straight x minus 1 equals tan squared straight x space to space simplify space further comma space and space then space integrate.
end style

Answered by Expert 20th March 2017, 6:37 PM
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