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General procedure for
integrating a fractional
rational function
(rational fraction)

1
...

2
...


3
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4
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5
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Recommendations for the
integration of the simplest
rational fractions
...


Mx  N
I2   2
dx
n
( x  px  q )
The denominator must be a full
square
...
The fraction is irregular, so
we represent it as a sum of a
polynomial and a regular
fraction, dividing the
numerator by a denominator
...
Decompose the
denominator of a regular
rational fraction to prime
factors
...
Imagine the regular
rational fraction as the sum
of the simplest rational
fractions:
A B
Cx  D
f 2 ( x)   2  2
x x
x  2x  2

4
...

4x  4x  4x  4

4
3
2
x  2x  2x A B
Cx  D
  2 2
x x
x  2x  2
3
2
4x  4x  4x  4 
3

2

 Ax (x  2x  2) 
2
 B(x  2x  2) 
2
 (Cx  D)x
2

4x  4x  4x  4 
2
 Ax (x  2x  2) 
2
 B(x  2x  2) 
2
 (Cx  D)x
3

2

Compare the coefficients for the
same degrees:
3
C4
at x 4  A  C
2
D2
at x 4  2 A  B  D
1
A0
at x 4  2A  2B
0
B2
at x 4  2B

Subintegral function takes
the form:

2
4x  2
f ( x )  ( x  2)  2  2
x
x  2x  2

5
...

2



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