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If m is a non-zero number and integral subscript blank superscript blank fraction numerator x to the power of 5 m minus 1 end exponent plus 2 x to the power of 4 m minus 1 end exponent over denominator left parenthesis x to the power of 2 m end exponent plus x to the power of m plus 1 right parenthesis cubed end fraction equals f left parenthesis x right parenthesis plus c, then f(x) is:

  1. fraction numerator x to the power of 5 m end exponent over denominator 2 m left parenthesis x to the power of 2 m end exponent plus x to the power of m end exponent plus 1 right parenthesis to the power of 2 end exponent end fraction    
  2. fraction numerator x to the power of 4 m end exponent over denominator 2 m left parenthesis x to the power of 2 m end exponent plus x to the power of m end exponent plus 1 right parenthesis to the power of 2 end exponent end fraction    
  3. fraction numerator 2 m left parenthesis x to the power of 5 m end exponent plus x to the power of 4 m end exponent right parenthesis over denominator left parenthesis x to the power of 2 m end exponent plus x to the power of m end exponent plus 1 right parenthesis to the power of 2 end exponent end fraction    
  4. fraction numerator left parenthesis x to the power of 5 m end exponent minus x to the power of 4 m end exponent right parenthesis over denominator 2 m left parenthesis x to the power of 2 m end exponent plus x to the power of m end exponent plus 1 right parenthesis to the power of 2 end exponent end fraction    

The correct answer is: fraction numerator x to the power of 4 m end exponent over denominator 2 m left parenthesis x to the power of 2 m end exponent plus x to the power of m end exponent plus 1 right parenthesis to the power of 2 end exponent end fraction


    I equals not stretchy integral subscript blank superscript blank fraction numerator x to the power of 5 m minus 1 end exponent plus 2 x to the power of 4 m minus 1 end exponent over denominator x to the power of 6 m end exponent left parenthesis 1 plus x to the power of negative m end exponent plus x to the power of negative 2 m end exponent right parenthesis to the power of 3 end exponent end fraction d x
    Put 1 plus x to the power of negative m end exponent plus x to the power of negative 2 m end exponent equals r
    -m x to the power of negative m minus l end exponent plus negative 2 m x to the power of negative 2 m minus 1 end exponent d x equals d r
    rightwards double arrow left parenthesis x to the power of negative m end exponent plus 2 x to the power of 2 m minus 1 end exponent right parenthesis d x equals fraction numerator negative 1 over denominator m end fraction d r rightwards double arrow fraction numerator negative 1 over denominator m end fraction not stretchy integral subscript blank superscript blank fraction numerator 1 over denominator r to the power of 3 end exponent end fraction d r
    equals fraction numerator negative 1 over denominator m end fraction left square bracket fraction numerator 1 over denominator 2 t to the power of 2 end exponent end fraction right square bracket equals fraction numerator 1 over denominator 2 m end fraction left parenthesis fraction numerator 1 over denominator left parenthesis 1 plus x to the power of negative m end exponent plus x to the power of negative 2 m end exponent right parenthesis to the power of 2 end exponent end fraction right parenthesis
    equals fraction numerator x to the power of 4 m end exponent over denominator 2 m left parenthesis x to the power of 2 m end exponent plus x to the power of m end exponent plus 1 right parenthesis to the power of 2 end exponent end fraction

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