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Question

If ellr means (log log log......), the log being repeated r times, then
not stretchy integral left curly bracket x l left parenthesis x right parenthesis l to the power of 2 end exponent left parenthesis x right parenthesis l to the power of 3 end exponent left parenthesis x right parenthesis.... l to the power of r end exponent left parenthesis x right parenthesis right curly bracket to the power of negative 1 end exponent d xis equal to

  1. l to the power of r plus 1 end exponent left parenthesis x right parenthesis plus C    
  2. fraction numerator l to the power of r plus 1 end exponent left parenthesis x right parenthesis over denominator r plus 1 end fraction+ c    
  3. l to the power of r end exponent left parenthesis x right parenthesis plus c    
  4. None of these    

The correct answer is: l to the power of r plus 1 end exponent left parenthesis x right parenthesis plus C


    To find the value of not stretchy integral left curly bracket x l left parenthesis x right parenthesis l to the power of 2 end exponent left parenthesis x right parenthesis l to the power of 3 end exponent left parenthesis x right parenthesis.... l to the power of r end exponent left parenthesis x right parenthesis right curly bracket to the power of negative 1 end exponent d x.
    l e t space l to the power of r plus 1 end exponent equals t
fraction numerator 1 over denominator x l left parenthesis x right parenthesis l squared left parenthesis x right parenthesis... l to the power of r left parenthesis x right parenthesis end fraction d x equals d t

integral fraction numerator 1 over denominator x l left parenthesis x right parenthesis l squared left parenthesis x right parenthesis... l to the power of r left parenthesis x right parenthesis end fraction d x equals integral 1 d t space equals t plus C

I equals l to the power of r plus 1 end exponent plus C

    Therefore, the value of  not stretchy integral left curly bracket x l left parenthesis x right parenthesis l to the power of 2 end exponent left parenthesis x right parenthesis l to the power of 3 end exponent left parenthesis x right parenthesis.... l to the power of r end exponent left parenthesis x right parenthesis right curly bracket to the power of negative 1 end exponent d x is l to the power of r plus 1 end exponent left parenthesis x right parenthesis plus C

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