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Integral calculus
  Integrating rational functions
 The Ostrogradsky method of the integration of a proper rational functions
 The integration of a rational fraction whose denominator Q (x) has multiple or repeated roots.
If P (x) and Q (x) are polynomials with real coefficients and P (x) / Q (x) is a proper fraction, and Q (x) has multiple roots, then
where Q1(x) is the greatest common divisor of Q (x) and its derivative Q' (x), while
Q2(x) = Q (x) / Q1(x).
Undetermined coefficients of he polynomials P1(x) and P2(x), whose degrees are one less than of the polynomials Q1(x) and Q2(x) respectively, we calculate by deriving the above integral identity.
 The Ostrogradsky method of the integration of a proper rational functions examples
 Example:  Evaluate
Since Q(x) = (x3 + 1)2  and  Q' (x) = 6x2(x3 + 1) then  Q1(x) = x3 + 1 is 
the greatest common divisor of  Q(x) and Q' (x), and  Q2(x) = Q (x) / Q1(x) = x3 + 1. Therefore,
 Solution:

 Example:  Evaluate
Since  Q(x) = (x + 1)2   (x2 + 1)2  and  Q' (x) = 2(x + 1)  (x2 + 1)2 + 4x(x + 1)2  (x2 + 1) or
               Q' (x) = 2(x + 1)(x2 + 1)[(x2 + 1) + 2x(x + 1)]  then  Q1(x) = (x + 1) (x2 + 1)   and
                Q2(x) = Q (x) / Q1(x) = (x + 1) (x2 + 1). Therefore,
 Solution:
 
 
 
 
 
 
 
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