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integral subscript 0 superscript straight infinity   open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis

  1. fraction numerator 1 over denominator log invisible function application a end fraction minus fraction numerator 1 over denominator log invisible function application b end fraction
  2. log invisible function application a minus log invisible function application b
  3. log invisible function application a plus log invisible function application b
  4. fraction numerator 1 over denominator log invisible function application a end fraction plus fraction numerator 1 over denominator log invisible function application b end fraction

hintHint:

Integrate the every term of the given expression.

The correct answer is: fraction numerator 1 over denominator log invisible function application a end fraction minus fraction numerator 1 over denominator log invisible function application b end fraction


    Given That:
    integral subscript 0 superscript straight infinity   open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis
    >>> Integrating every term gives :
    integral subscript 0 superscript straight infinity   open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis
    integral subscript 0 superscript infinity left parenthesis a to the power of negative x end exponent right parenthesis space minus space integral subscript 0 superscript infinity left parenthesis b to the power of negative x end exponent right parenthesis
    fraction numerator 1 over denominator log a end fraction minus fraction numerator 1 over denominator log b end fraction.
    >>> Therefore, the integration of integral subscript 0 superscript straight infinity   open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis is fraction numerator 1 over denominator log a end fraction minus fraction numerator 1 over denominator log b end fraction.

    Therefore, the integration of integral subscript 0 superscript straight infinity   open parentheses straight a to the power of negative straight x end exponent minus straight b to the power of negative straight x end exponent close parentheses dx equals left parenthesis straight a greater than 1 comma straight b greater than 1 right parenthesis is fraction numerator 1 over denominator log a end fraction minus fraction numerator 1 over denominator log b end fraction.

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