Question

If $straight f left parenthesis straight x right parenthesis space equals space fraction numerator straight e to the power of straight x over denominator 1 plus straight e to the power of straight x end fraction. space straight I subscript 1 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space xg open curly brackets straight x open parentheses 1 minus straight x close parentheses close curly brackets dx space and space straight I subscript 2 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space straight g open curly brackets straight x open parentheses 1 minus straight x close parentheses close curly brackets dx$ then the value of I₂/I₁ is

2

-2

1

-1

Solution

f(-a) + f(a) = 1
$straight I subscript 1 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space xg left curly bracket straight x left parenthesis 1 minus straight x right parenthesis right curly bracket space dx space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space left parenthesis 1 minus straight x right parenthesis space straight g space left curly bracket space straight x space left parenthesis 1 minus straight x right parenthesis right curly bracket dx open parentheses because space integral subscript straight a superscript straight b space straight f space left parenthesis straight x right parenthesis space dx space equals space integral subscript straight a superscript straight b space straight f space left parenthesis straight a plus straight b minus straight x right parenthesis space dx close parentheses 2 straight I subscript 1 space equals space integral subscript straight f left parenthesis negative straight a right parenthesis end subscript superscript straight f left parenthesis straight a right parenthesis end superscript space straight g left curly bracket straight x space left parenthesis 1 minus straight x right parenthesis right curly bracket space dx space equals space straight I subscript 2 rightwards double arrow space straight I subscript 2 divided by straight I subscript 1 space space equals space 2$

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