Simplifying and Integrating Complex Rational Functions

In this article, we will explore the process of integrating complex rational functions, using an example to demonstrate the steps involved in simplifying and integrating these functions. Understanding the principles behind polynomial long division and partial fraction decomposition is crucial for solving such problems.

Introduction to Integration of Rational Functions

Rational functions, which are quotients of polynomials, often need to be simplified and integrated. This involves several steps, including factoring, polynomial long division, and partial fraction decomposition. The goal is to express a complex rational function in a simpler form that can be easily integrated.

Example Problem: Integrating a Rational Function

Let us consider the integral:

I ∫ (5x^226)/(x^32x^2x) dx

Step 1: Simplify the Denominator

First, we simplify the denominator. The denominator can be factored as:

x^3 - 2x^2 - x x * (x - 1)^2

Step 2: Rewrite the Integral

Now, we can rewrite the integral as:

I ∫ (5x^2 26)/(x * (x - 1)^2) dx

Step 3: Perform Polynomial Long Division

Since the degree of the numerator (2) is less than the degree of the denominator (3), we can perform polynomial long division.

(5x^2 26) / (x * (x - 1)^2) 5 (31x - 26)/(x * (x - 1)^2)

Step 4: Partial Fraction Decomposition

Next, we decompose the remainder into partial fractions:

(31x - 26)/(x * (x - 1)^2) A/x B/(x - 1) C/(x - 1)^2

Step 5: Solve for Constants

Multiplying through by the denominator, we get:

31x - 26 A(x - 1)^2 Bx(x - 1) Cx

Expanding and collecting like terms, we find the constants:

A B 31

-2A - B C -26

A 6

Substituting A 6 into the first equation:

6 B 31 implies B 25

Substituting A 6 and B 25 into the second equation:

-12 - 25 C -26 implies C 9

Step 6: Write the Partial Fractions

Thus, we have:

(31x - 26)/(x * (x - 1)^2) 6/x 25/(x - 1) 9/(x - 1)^2

Step 7: Integrate Each Term

Now we can integrate each term separately:

I ∫ (5 6/x 25/(x - 1) 9/(x - 1)^2) dx

5x 6 ln|x| - 25 ln|x - 1| - 9/(x - 1) C

Final Result

Thus, the integral is:

∫ (5x^226)/(x^32x^2x) dx 5x - 25 ln|x - 1| - 9/(x - 1) C

Conclusion

Mastering the process of integrating complex rational functions involves a combination of polynomial long division and partial fraction decomposition. By following the steps systematically, one can solve these problems effectively and efficiently.