A rational function which I forget how to integrate

[Damascus Road]

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, \int \frac{x}{x^{4}-1} dx


Thanks, in advance!

 

[Mark44]

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.

 

[gabbagabbahey]

That's not a very good example...the denominator can be factored.

edit Mark beat me to it

 

[Damascus Road]

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!

 

[gabbagabbahey]

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.