Integrating Some Rational Functions

​
numerator coefficients
n
0
-2
n
1
3
n
2
0
number of denominator roots
d
2
3
denominator roots
d
1
1
d
2
-1
d
3
0
help
method
1
2
show solution of the system
show calculated integral
Find the indefinite integral:

-2+3x
(-1+x)x(1+x)
x
Find the coefficients in the partial fraction decomposition first:
3x-2
(x-1)x(x+1)
=
a
x-1
+
b
x+1
+
c
x
Multiply the above equation by the common denominator to get:
3x-2 = a(x+1)x+b(x-1)x+c(x-1)(x+1)
Equate coefficients of powers of
x
and solve the system of linear equations:
1
-2-c
x
3a-b
2
x
0a+b+c
a
1
2
b
-
5
2
c
2
Integrate the rational functions in the decomposition.
1
2
log(x-1)+2log(x)-
5
2
log(x+1)
This Demonstration shows how to integrate a rational function whose denominator is a polynomial of degree 2 or 3 with real roots. The function is decomposed into partial fractions, which are then integrated. Two methods are used to determine the coefficients in the decomposition. The first consists of equating coefficients of powers of
x
on the left and right sides of the corresponding equation. In the second method, the roots of the denominator are substituted for
x
. In the case of multiple roots, some other value should be substituted.

External Links

Partial Fraction Decomposition (Wolfram MathWorld)
Partial Fraction Decomposition
Rational Function (Wolfram MathWorld)
Indefinite Integral (Wolfram MathWorld)
Find the Coefficients of a Partial Fraction Decomposition
Integrating a Quadratic Divided by the Square Root of a Quadratic

Permanent Citation

Izidor Hafner, Ed Pegg Jr
​
​"Integrating Some Rational Functions"​
​http://demonstrations.wolfram.com/IntegratingSomeRationalFunctions/​
​Wolfram Demonstrations Project​
​Published: April 30, 2014