Question

$Prove space that space straight d over dx open curly brackets fraction numerator 1 over denominator 4 square root of 2 end fraction log fraction numerator straight x squared plus straight x square root of 2 plus 1 over denominator straight x squared minus straight x square root of 2 plus 1 end fraction plus fraction numerator 1 over denominator 2 square root of 2 end fraction tan to the power of negative 1 end exponent fraction numerator straight x square root of 2 over denominator 1 minus straight x squared end fraction close curly brackets equals fraction numerator 1 over denominator 1 plus straight x to the power of 4 end fraction$

Solution

straight L. straight H. straight S. equals straight d over dx open curly brackets fraction numerator 1 over denominator 4 square root of 2 end fraction log fraction numerator straight x squared plus straight x square root of 2 plus 1 over denominator straight x squared minus straight x square root of 2 plus 1 end fraction plus fraction numerator 1 over denominator 2 square root of 2 end fraction tan to the power of negative 1 end exponent fraction numerator straight x square root of 2 over denominator 1 minus straight x squared end fraction close curly brackets equals straight d over dx open curly brackets fraction numerator 1 over denominator 4 square root of 2 end fraction log left parenthesis straight x squared plus straight x square root of 2 plus 1 right parenthesis minus log left parenthesis straight x squared minus straight x square root of 2 plus 1 right parenthesis plus fraction numerator 1 over denominator 2 square root of 2 end fraction tan to the power of negative 1 end exponent fraction numerator straight x square root of 2 over denominator 1 minus straight x squared end fraction close curly brackets equals fraction numerator 1 over denominator 4 square root of 2 end fraction open square brackets fraction numerator 2 straight x plus square root of 2 over denominator straight x squared plus straight x square root of 2 plus 1 end fraction minus fraction numerator 2 straight x minus square root of 2 over denominator straight x squared minus straight x square root of 2 plus 1 end fraction close square brackets plus fraction numerator 1 over denominator 2 square root of 2 end fraction fraction numerator 1 over denominator 1 plus open parentheses begin display style fraction numerator straight x square root of 2 over denominator 1 minus straight x squared end fraction end style close parentheses squared end fraction. fraction numerator left parenthesis 1 minus straight x squared right parenthesis. square root of 2 minus straight x square root of 2 left parenthesis negative 2 straight x right parenthesis over denominator open parentheses 1 minus straight x squared close parentheses squared end fraction equals fraction numerator 1 over denominator 4 square root of 2 end fraction open square brackets fraction numerator 2 straight x cubed minus 2 square root of 2 straight x squared plus 2 straight x plus square root of 2 straight x squared minus 2 straight x plus square root of 2 minus 2 straight x cubed minus 2 square root of 2 straight x squared minus 2 straight x plus square root of 2 straight x squared plus 2 straight x plus square root of 2 over denominator left parenthesis straight x squared plus straight x square root of 2 plus 1 right parenthesis left parenthesis straight x squared minus straight x square root of 2 plus 1 right parenthesis end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space plus fraction numerator 1 over denominator 2 square root of 2 end fraction fraction numerator left parenthesis 1 minus straight x squared right parenthesis squared over denominator left parenthesis 1 minus straight x squared right parenthesis squared plus 2 straight x squared end fraction cross times fraction numerator square root of 2 minus square root of 2 straight x squared plus 2 square root of 2 straight x squared over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction equals fraction numerator 1 over denominator 4 square root of 2 end fraction open square brackets fraction numerator 2 square root of 2 minus 2 square root of 2 straight x squared over denominator left parenthesis straight x squared plus 1 right parenthesis squared minus 2 straight x squared end fraction close square brackets plus fraction numerator 1 over denominator 2 square root of 2 end fraction fraction numerator square root of 2 plus square root of 2 straight x squared over denominator left parenthesis 1 minus straight x squared right parenthesis squared plus 2 straight x squared end fraction equals 1 half cross times fraction numerator 1 minus straight x squared over denominator 1 plus straight x to the power of 4 end fraction plus 1 half cross times fraction numerator 1 plus straight x squared over denominator 1 plus straight x to the power of 4 end fraction equals fraction numerator 1 over denominator 2 left parenthesis 1 plus straight x to the power of 4 right parenthesis end fraction left square bracket 1 minus straight x squared plus 1 plus straight x squared right square bracket equals fraction numerator 1 over denominator 2 left parenthesis 1 plus straight x to the power of 4 right parenthesis end fraction cross times 2 equals fraction numerator 1 over denominator 1 plus straight x to the power of 4 end fraction equals straight R. straight H. straight S.

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