Integrating 1/(x+x^3+2): Simplified Steps & Techniques

[Clara Chung]

Homework Statement

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

Homework Equations

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]

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

 

[vela]

Try the substitution $u=x-2$ on the second integral.

 

[Clara Chung]

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]

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.)

 

[vela]

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]

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?