A rational function which I forget how to integrate

In summary, In order to integrate a rational function with an unfactorable denominator, one can use the method of partial fraction decomposition. This involves rewriting the rational function as a sum of simpler fractions, with the numerators being of degree one less than the denominators. The last term in the sum must have a first degree polynomial for the numerator, while the earlier terms can have a constant or a first degree polynomial depending on the degree of the corresponding denominator.
  • #1
Damascus Road
120
0
Hey all!

It's been a while since I've done this, how do you integrate a rational function, where the denominator cannot be factored, again?

For example, [tex]\int \frac{x}{x^{4}-1} dx [/tex]


Thanks, in advance!
 
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  • #2
The denominator can be factored, though. The method you're probably thinking of is called Partial Fraction Decomposition.

You want to rewrite x/(x^4 - 1) as A/(x - 1) + B/(x + 1) + (Cx + D)/(x^2 + 1).
Solve for A, B, C, and D.
 
  • #3
That's not a very good example:smile:...the denominator can be factored.

edit Mark beat me to it :cry:
 
  • #4
I should've seen that... my bad. However,
can you explain why you placed Cx + D above the irreducible part, instead of just "C"?

Thanks!
 
  • #5
Damascus Road said:
I should've seen that... my bad. However,
can you explain why you placed Cx + D above the irreducible part, instead of just "C"?

Thanks!

For the method of partial fraction decomposition, in general, the numerator of each term must be of degree one less than the degree of the corresponding denominator. The last term has a 2nd degree polynomial for the denominator, so you ,in general, require a first degree polynomial for the numerator.

P.S. the reason I wrote "in general" in the above statement, is that sometimes you will find C=0.
 

Related to A rational function which I forget how to integrate

1. What is a rational function?

A rational function is a mathematical expression that can be written as a ratio of two polynomials, where the denominator is not equal to zero. It is also known as a rational expression.

2. How do you integrate a rational function?

To integrate a rational function, you first need to use algebraic manipulation to rewrite it as a sum of simpler fractions. Then, you can use the power rule or substitution method to integrate each individual fraction.

3. What is the power rule?

The power rule is a basic integration rule that states that the integral of x^n is equal to (x^(n+1))/(n+1) + C, where C is a constant. This rule can be used to integrate monomial rational functions.

4. When can substitution be used to integrate a rational function?

Substitution can be used to integrate a rational function when the numerator can be expressed as the derivative of the denominator. This allows for the rational function to be rewritten as a simpler expression that can be easily integrated.

5. What are some common techniques for integrating more complex rational functions?

Some common techniques for integrating more complex rational functions include partial fraction decomposition, trigonometric substitution, and integration by parts. These methods can be used to break down the rational function into simpler parts that can be integrated using known techniques.

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