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If the primitive of text end text s i n to the power of negative 3 divided by 2 end exponent invisible function application x s i n to the power of negative 1 divided by 2 end exponent invisible function application left parenthesis x plus theta right parenthesis text end text text is end text text end text minus 2 c o s e c invisible function application theta square root of f left parenthesis x right parenthesis end root plus C text end textthen

  1. f left parenthesis x right parenthesis equals fraction numerator sin invisible function application x over denominator s i n invisible function application left parenthesis x plus theta right parenthesis end fraction    
  2. f left parenthesis x right parenthesis equals t a n invisible function application left parenthesis x plus theta right parenthesis    
  3. f left parenthesis x right parenthesis equals fraction numerator s i n invisible function application left parenthesis x plus theta right parenthesis over denominator sin invisible function application x end fraction    
  4. f left parenthesis x right parenthesis equals fraction numerator t a n invisible function application left parenthesis x plus theta right parenthesis over denominator tan invisible function application x end fraction    

The correct answer is: f left parenthesis x right parenthesis equals fraction numerator s i n invisible function application left parenthesis x plus theta right parenthesis over denominator sin invisible function application x end fraction


    I equals integral fraction numerator d x over denominator square root of s i n to the power of 3 end exponent invisible function application x left parenthesis s i n invisible function application x plus theta right parenthesis end root end fraction equals integral fraction numerator c o s e c to the power of 2 end exponent invisible function application x d x over denominator square root of fraction numerator s i n invisible function application left parenthesis x plus theta right parenthesis over denominator sin invisible function application x end fraction end root end fraction equals fraction numerator negative 1 over denominator sin invisible function application theta end fraction integral fraction numerator d t over denominator square root of t end fraction text  where  end text t equals fraction numerator s i n invisible function application left parenthesis x plus theta right parenthesis over denominator sin invisible function application x end fraction
    equals negative 2 c o s e c invisible function application theta square root of fraction numerator s i n invisible function application left parenthesis x plus theta right parenthesis over denominator sin invisible function application x end fraction plus C end root

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