Maths-
General
Easy

Question

The partial fractions of fraction numerator 6 x to the power of 4 end exponent plus 5 x to the power of 3 end exponent plus x to the power of 2 end exponent plus 5 x plus 2 over denominator 1 plus 5 x plus 6 x to the power of 2 end exponent end fraction are

  1. x to the power of 2 end exponent minus fraction numerator 1 over denominator 1 plus 2 x end fraction plus fraction numerator 1 over denominator 1 plus 3 x end fraction    
  2. x to the power of 2 end exponent plus fraction numerator 1 over denominator 1 plus 2 x end fraction plus fraction numerator 1 over denominator 1 plus 3 x end fraction    
  3. x to the power of 2 end exponent plus fraction numerator 1 over denominator 1 plus 2 x end fraction minus fraction numerator 1 over denominator 1 plus 3 x end fraction    
  4. negative x to the power of 2 end exponent plus fraction numerator 1 over denominator 1 plus 2 x end fraction plus fraction numerator 1 over denominator 1 plus 3 x end fraction    

hintHint:

In order to integrate a rational function, it is reduced to a proper rational function. The method in which the integrand is expressed as the sum of simpler rational functions is known as decomposition into partial fractions. After splitting the integrand into partial fractions, it is integrated accordingly with the help of traditional integrating techniques.
The stepwise procedure for finding the partial fraction decomposition is explained here::
  • Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.
  • Step 2: Now, factor the denominator of the rational expression into the linear factor or in the form of irreducible quadratic factors (Note: Don’t factor the denominators into the complex numbers).
  • Step 3: Write down the partial fraction for each factor obtained, with the variables in the numerators, say A and B.
  • Step 4: To find the variable values of A and B, multiply the whole equation by the denominator.
  • Step 5: Solve for the variables by substituting zero in the factor variable.
  • Step 6: Finally, substitute the values of A and B in the partial fractions.

The correct answer is: x to the power of 2 end exponent plus fraction numerator 1 over denominator 1 plus 2 x end fraction plus fraction numerator 1 over denominator 1 plus 3 x end fraction


     Given :
    fraction numerator 6 x to the power of 4 end exponent plus 5 x to the power of 3 end exponent plus x to the power of 2 end exponent plus 5 x plus 2 over denominator 1 plus 5 x plus 6 x to the power of 2 end exponent end fraction
    Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.
    rightwards double arrow space fraction numerator 6 x to the power of 4 plus 5 x cubed plus x squared plus 5 x plus 2 over denominator 1 plus 5 x plus 6 x squared end fraction space equals space fraction numerator 6 x to the power of 4 plus 5 x cubed plus x squared over denominator 1 plus 5 x plus 6 x squared end fraction space plus space fraction numerator 5 x plus 2 over denominator 1 plus 5 x plus 6 x squared end fraction

F u r t h e r space s i m p l i f y i n g

rightwards double arrow fraction numerator x squared left parenthesis 6 x squared plus 5 x plus 1 right parenthesis over denominator 1 plus 5 x plus 6 x squared end fraction space plus space fraction numerator 5 x plus 2 over denominator left parenthesis 2 x space plus 1 right parenthesis left parenthesis 3 x plus 1 right parenthesis end fraction space rightwards double arrow x squared space plus space fraction numerator 5 x plus 2 over denominator left parenthesis 2 x space plus 1 right parenthesis left parenthesis 3 x plus 1 right parenthesis end fraction

    Splitting 5x + 2 in the numerator
    rightwards double arrow x squared space plus space fraction numerator left parenthesis 3 x plus 1 right parenthesis space plus left parenthesis 2 x plus 1 right parenthesis over denominator left parenthesis 2 x space plus 1 right parenthesis left parenthesis 3 x plus 1 right parenthesis end fraction space

S p l i t t i n g space a n d space c a n c e l l i n g space l i k e space f a c t o r s
rightwards double arrow x squared space plus space fraction numerator 1 over denominator left parenthesis 2 x space plus 1 right parenthesis end fraction space plus space fraction numerator 1 over denominator left parenthesis 3 x space plus 1 right parenthesis end fraction space

    Thus, the partial fractions of fraction numerator 6 x to the power of 4 end exponent plus 5 x to the power of 3 end exponent plus x to the power of 2 end exponent plus 5 x plus 2 over denominator 1 plus 5 x plus 6 x to the power of 2 end exponent end fraction are  x squared space plus space fraction numerator 1 over denominator left parenthesis 2 x space plus 1 right parenthesis end fraction space plus space fraction numerator 1 over denominator left parenthesis 3 x space plus 1 right parenthesis end fraction space

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