←U-substitution with a definite integralIntegration

4.4.4 Integration using partial fraction decomposition

Partial fraction decomposition is used to prepare a rational function for integration. A rational function is the quotient of two polynomials: f(x)=g(x)÷h(x). Partial-fraction decomposition changes a rational function into a sum of fractions with simpler denominators. Myron has a transformation to aid in the application of partial-fraction decomposition.

To integrate a rational function, four major steps are performed.

  1. Reduce the rational; this is only performed when the dividend is a polynomial of higher degree than that of the divisor .
  2. Decompose the denominator into irreducible linear and quadratic factors.
  3. Find a partial-fraction decomposition of the factors. (See §3.4.9.)
  4. Finally, integrate the decomposition.

The steps for an example are summarized in Figure 4.1. Column b contains expressions resulting from transformations on expressions in column a.

Step a. b.
1 ∫(x^5+2)÷(x^2-1) ⅆx ∫((x^2-1)⋅(x^3+x)+(x+2))÷(x^2-1) ⅆx
1.1 ∫x^3+x+(x+2)÷(x^2-1) ⅆx
2 ∫x^3+x+(x+2)÷(x^2-1) ⅆx ∫x^3+x+(x+2)÷((x-1)⋅(x+1)) ⅆx
3 (x+2)÷((x-1)⋅(x+1)) (x+2)÷((x-1)⋅(x+1))=A÷(x-1)+B÷(x+1)
3.1 x+2=(x-1)⋅(x+1)⋅(A÷(x-1)+B÷(x+1))
3.2 x+2=(x+1)⋅A+(x-1)⋅B
3.3 x+2=x⋅(A+B)+A-B
3.4 A+B=1; A-B=2
3.5 B=-1/2; A=3/2 (x+2)÷((x-1)⋅(x+1))=3/2÷(x-1)-1/2÷(x+1)
4 ∫x^3+x+3/2÷(x-1)-1/2÷(x+1) ⅆx x^4÷4+x^2÷2+3/2⋅ln |x-1|-1/2⋅ln |x+1|
Figure 4.1 Steps in fractional decomposition

The summary above is considerably condensed. Here are the missing details.

1.b Perform Polynomial Division on the integrand to reduce the rational.

1.1b Distribute the fraction, Simplify , collect terms with the same denominator and Simplify again.

2.b Reduce x^2-1 by finding its roots using Quadratic and Simplify .

3.b Select the rational function as a subexpression of the integral and apply Partial Fraction .

3.1.b In the equation introduced by 3.b, rearrange the denominator on the left to be a multiplicand on the right.

3.2.b Distribute the new multiplicand across the former right side.

3.3.b Distribute the expression on the right. Collect terms in x and factor.

3.4.a Use selection and partial duplication, , along with + to create the equations A+B=1 and A-B=2. Isolate A and B, substitute and solve.

3.5.b Substitute A and B into 3.b

4.a Select the rational function in 2.b and substitute from the right side of 3.5.b.

4.b Transform the integral using symbolic simplification.