7.3. Integration by Partial Fractions http://www.ck12.org
2 x− 19
x^2 +x− 6 =A
x+ 3 +B
x− 2.Our goal at this point is to find the values ofAandB. To solve this equation, multiply both sides of the equation by
the factored denominator(x+ 3 )(x− 2 ).This process will produce thebasic equation.
2 x− 19 =A(x− 2 )+B(x+ 3 ).This equation is true for all values ofx.The most convenient values are the ones that make a factor equal to zero,
namely,x=2 andx=− 3 .Substitutingx= 2 ,
2 ( 2 )− 19 =A( 2 − 2 )+B( 2 + 3 )
− 15 = 0 + 5 B
− 3 =B
Similarly, substituting forx=−3 into the basic equation we get
2 (− 3 )− 19 =A(− 3 − 2 )+B(− 3 + 3 )
− 25 =− 5 A+ 0
5 =A
We have solved the basic equation by finding the values ofAandB.Therefore, the partial fraction decomposition is
2 x− 19
x^2 +x− 6 =5
x+ 3 −3
x− 2.General Description of the Method
To be able to write a rational functionf(x)/g(x)as a sum of partial fractions, we must apply two conditions:
- The degree off(x)must be less than the degree ofg(x).If so, the rational function is calledproper. If it is
not, dividef(x)byg(x)(use long division) and work with the remainder term. - The factors ofg(x)are known. If not, you need to find a way to find them. The guide below shows how you
can writef(x)/g(x)as a sum of partial fractions if the factors ofg(x)are known.
A Guide to Finding Partial Fractions Decomposition of a Rational Function
- To find the partial fraction decomposition of a proper rational function,f(x)/g(x),factor the denominatorg(x)
and write an equation that has the form
f(x)
g(x)= (sum of partial fractions.)