Is there a general rule for the evaluation of a rational polynomial function?

[Al-Layth]

[math]\int {\frac{p(x)}{q(x)}}dx =?[/math]
where p(x) and q(x) and polynomials.

is there a rule to evaluate this integral in a closed form (no modifications, just a result in terms of p(x) and q(x)?

 

[Otis]

is there a rule to evaluate this integral in a closed form … in terms of p(x) and q(x)?

Hi. In math, the verb 'evaluate' means to find a numerical value. You may be thinking about finding an antiderivative, instead. If so, then the answer is 'no', not in terms of variables p(x) and q(x). <edited>

We would first need to express functions p and q as polynomials. One way to do that is by partial fraction decomposition, as shown in the following thread.

Is there a general formula for the antiderivative of rational functions?

Some antiderivatives of rational functions involve inverse trigonometric functions, and some involve logarithms. But inverse trig functions can be expressed in terms of complex logarithms. So is th...

math.stackexchange.com

[imath]\;[/imath]

 

[Al-Layth]

Hi. In math, the verb 'evaluate' means to find a numerical value. You may be thinking about finding an antiderivative, instead. If so, then the answer is 'no'.

We would first need to express functions p and q as polynomials. One way to do that is by partial fraction decomposition, as shown in the following thread.

Is there a general formula for the antiderivative of rational functions?

Some antiderivatives of rational functions involve inverse trigonometric functions, and some involve logarithms. But inverse trig functions can be expressed in terms of complex logarithms. So is th...

math.stackexchange.com

[imath]\;[/imath]

you said no but the site u linked to provides an antiderivative for it <_<

 

[NotJeffM]

I note that the linked page says that an anti-derivative exists and that it depends on factoring the denominator into linear terms. I am not sure that knowing that the anti-derivative exists tells you how to determine it. Every polynomial of degree n can be factored into n linear factors. I have never found that fact to be very helpful in finding a factorization.

 

[Steven G]

If deg(p(x)) > deg(q(x)) you should 1st do long division. That is you really should be asking if there is a formula for the integral you described if deg(p(x)) < deg(q(x))

 

[Otis]

you said no

I meant that there's no formula using variable symbols p(x) and q(x), as you'd requested. Please excuse.

Do you understand what they've done at that link?
[imath]\;[/imath]

 

[Al-Layth]

I meant that there's no formula using variable symbols p(x) and q(x), as you'd requested. Please excuse.

Do you understand what they've done at that link?
[imath]\;[/imath]

I understood 100% of the 50% i understood

 

[Al-Layth]

If deg(p(x)) > deg(q(x)) you should 1st do long division. That is you really should be asking if there is a formula for the integral you described if deg(p(x)) < deg(q(x))

yes i forgot that condition