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Question

integral x to the power of 3 end exponent square root of fraction numerator 1 plus x to the power of 2 end exponent over denominator 1 minus x to the power of 2 end exponent end fraction end root d x equals A square root of 1 minus x to the power of 4 end exponent end root minus B x to the power of 2 end exponent square root of 1 minus x to the power of 4 end exponent end root plus C s i n to the power of negative 1 end exponent invisible function application x to the power of 2 end exponent plus D comma text end textthen A+B+C is equal to

  1. negative fraction numerator 1 over denominator 2 end fraction    
  2. negative fraction numerator 1 over denominator 4 end fraction    
  3. fraction numerator 1 over denominator 4 end fraction    
  4. fraction numerator 1 over denominator 2 end fraction    

The correct answer is: negative fraction numerator 1 over denominator 2 end fraction


    fraction numerator 1 over denominator 2 end fraction integral fraction numerator square root of 1 plus x to the power of 2 end exponent end root over denominator square root of 1 minus x to the power of 2 end exponent end root end fraction x to the power of 2 end exponent left parenthesis 2 x right parenthesis d x
    text Let  end text x to the power of 2 end exponent equals t equals fraction numerator 1 over denominator 2 end fraction integral fraction numerator square root of 1 plus t end root over denominator square root of 1 minus t end root end fraction times t d t equals fraction numerator 1 over denominator 2 end fraction integral fraction numerator t plus t to the power of 2 end exponent over denominator square root of 1 minus t to the power of 2 end exponent end root end fraction d t
    not break into 2
    A equals negative fraction numerator 1 over denominator 2 end fraction comma B equals negative fraction numerator 1 over denominator 4 end fraction comma C equals fraction numerator 1 over denominator 4 end fraction

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