Can I solve integrals and limits without using L'Hopital's rule?

[sutupidmath]

integral, and limit help?

Homework Statement

i am terrible with latex, so i will just write it down.

integ of 1/(x^3+x^2+1)dx

and the limit i am trying to do it without using the l'hopital rule. By the way i also would like to know if there exists any theorem which states that, if the limit of a functions can be calculated using l'hopital rule, than it will be solvable also without using l'hopital rule??

lim{x-->0)(x cos(x)-sin(x) )/( x- sin(x) )

using l'hopital rule, the limit is -2

thnx in advance

Homework Equations

The Attempt at a Solution

 

[cristo]

Is this two different questions, one integral and one limit? You've not really made that very clear! For the integral, what have you tried. For the limit, why do you want to not use L'Hopital?

 

[sutupidmath]

yeah, it is two different questions.
as long as i don't know how to use latex i think that there would be to much symbols if i posted all i have tried so far. However i am just going to summarize in words what i have already done. I have a feeling that i have to try to form some kind of a perfect square of two expressions, like A^2+ b^2, where the first one should contain the variable x, and the latter (b^2) should be some constant. But so far i have not been able to do this.
Another thing i have tried is based on the fact that if we substituted x^3+x^2+1=t,then when we differentiate both sides we get, (2x^2+2x)dx=dt, so in order to be able to apply such a substitution i need to have the same expression (2x^2+2x)dx, at the numerator, but i don't know how to go about forming it.
Another thing i have tried is to take the substitution x=tan(t/2), and then sinx=(2t)/(1+t^2), and also the proper expression for the cosine. But again have gotten nothing.

--- As for the limit, as i stated i was just wondering if it could be done without using the l'hopital rule?

thnx

 

[sutupidmath]

is anyone out there going to give me some hints on this integral??

 

[Dick]

I think the only way you could do something reasonable with that integral is to factor the denominator and use partial fractions. But you can't. It doesn't have any rational roots.

 

[sutupidmath]

I think the only way you could do something reasonable with that integral is to factor the denominator and use partial fractions. But you can't. It doesn't have any rational roots.

yeah i know i cannot factor the denominator, since it does not have any roots over reals. But i thought i could use some tricks like we do with the integral:

integ dx/(x^2+x+1)

the denominator here also does not have any roots over reals, however it is pretty easy after we play some tricks with the denominator.
So i had a feeling that i could do something like this also with the

integ dx/(x^3+x^2+1) but so far have gotten nowhere.

 

[HallsofIvy]

Yes, x²+ x+ 1 has no real roots- but it is already quadratic so you can complete the square.
The denominator x³+ x²+ 1 does have a real root- any odd degree polynomial must have at least one. But it has no rational roots so you aren't going to be able to factor it easily. Unfortunately, "partial fractions" requires that all factors be quadratic or linear and you must know at least one root to do that.