Integral calculus
     The indefinite integral
      Integrating rational functions
         The Ostrogradsky method of the integration of a proper rational functions
         The Ostrogradsky method of the integration of a proper rational functions, examples
Integrating rational functions
Recall that a rational function is a ratio of two polynomials written  Pn(x) / Qm(x).
To integrate a rational function
in case n > m, that is, if the degree of P is greater than or equal to degree of Q, we first divide Pn(x) by Qm(x) to obtain the sum of the polynomial Pn -  m(x) and a proper or simple rational function.
Then, we integrate proper rational function using decomposition of rational function into a sum of partial fractions.
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:  72. 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:  73. 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: 
Calculus contents F
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