Integrate 1/(x+x^3+2)

Homework Statement

integrate 1/(x+x^3+2)

The Attempt at a Solution

I have tried to use partial fraction but the process is very complicated. Are there any faster methods? It is one of the ten questions in a 50 minutes elementary level test.

Here is my attempt,
∫ 1/(x^3+x+2) dx= ∫ 1/[4(x+1)] dx - ∫ (x-2)/[4(x^2-x+2)] dx
= In(x+1)/4 - ∫ (x-2)/[4((x-1/2)^2+7/4)] dx + C
Then I have to use complicated trigonometry by letting x - 1/2 = (root 7) tan(y) /2

Please tell me if there is a faster method

Ackbach
Gold Member
Don't think there's a faster method. Push forward with what you have!

vela
Staff Emeritus
Homework Helper
Try the substitution ##u=x-2## on the second integral.

Ackbach
Try the substitution ##u=x-2## on the second integral.
Please teach me how to do after letting u = x-2
∫ (x-2)/[(x^2-x+2)] dx
= ∫ (u / (u^2 + 3u + 4)) du
=???

Ray Vickson
Homework Helper
Dearly Missed
Please teach me how to do after letting u = x-2
∫ (x-2)/[(x^2-x+2)] dx
= ∫ (u / (u^2 + 3u + 4)) du
=???

What do YOU think you should do next? What have you tried so far? (I mean besides the ##x\; \text{to}\;y## transformation you already mentioned and that you said you would like to avoid.)

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Ackbach
vela
Staff Emeritus
Homework Helper
Please teach me how to do after letting u = x-2
∫ (x-2)/[(x^2-x+2)] dx
= ∫ (u / (u^2 + 3u + 4)) du
=???
I made a mistake when thinking of how to do the integral, so I don't think this substitution really helps. As Ackbach advised earlier, just carry on with your initial attempt.

epenguin
epenguin
Homework Helper
Gold Member
So so you are doing it by stages and the second stage getting something like

$$\int \dfrac {\left( x-2\right) dx}{\left[ 4\left( x-\dfrac {1}{2}\right) ^{2}+\dfrac {7}{4}\right] }$$

(I have not checked and am not sure of the placing of the 4 and brackets from your text, but the following holds in any case).

So that is a linear (first degree) function divided by a quadratic (second degree). For the next (I am afraid not the last) step - in calculus what is the relation between some first and a second degree polynomial?

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