## LACHEEK989 Group Title ∫1/x2+x+1 one year ago one year ago

1. wio Group Title

Find the roots, then you can factor it. Then you want to try partial fraction decomposition.

2. LACHEEK989 Group Title

I am lost at completing the square.

3. zepdrix Group Title

$\large x^2+bx$To complete the square, we take half of the $$\large b$$ term, and square it. $\large x^2+bx+\left(\frac{b}{2}\right)^2$Now we can't just add this number, it will change the value of our polynomial, we want to keep it balanced. So we have to also subtract this number. $\large \color{royalblue}{x^2+bx+\left(\frac{b}{2}\right)^2}-\left(\frac{b}{2}\right)^2$The reason for doing this is, the blue part will now factor down into a perfect square. A binomial containing x and half of the b term.$\large \color{royalblue}{\left(x+\frac{b}{2}\right)^2}-\left(\frac{b}{2}\right)^2$

4. zepdrix Group Title

Let's see if we can apply this to our problem here.

5. zepdrix Group Title

$\large x^2+x+1$The $$\large b$$ coefficient appears to be 1. (The middle term ~ because $$\large x$$ is the same as $$\large 1\cdot x$$ ). So to to complete the square, we'll take half of 1, and square it.$\large \color{orangered}{x^2+x+\left(\frac{1}{2}\right)^2}-\left(\frac{1}{2}\right)^2+1$I moved the 1 out of the way, we don't want to deal with that right now. Ok the orange part should give us a perfect square now,$\large \color{orangered}{\left(x+\frac{1}{2}\right)^2}-\left(\frac{1}{2}\right)^2+1$ Which will simplify further to,$\large \left(x+\frac{1}{2}\right)^2+\frac{3}{4}$

6. LACHEEK989 Group Title

$4/3 \int\limits_{}^{}1/((4/3) u^2 +1) du$ What next?

7. LACHEEK989 Group Title

(3/4)∫1/((4/3)u2+1)du I think this is right. What do I do next?

8. Outkast3r09 Group Title

this seems much more easier than what is being done... but thats just me

9. Outkast3r09 Group Title

$\frac{4}{3}\int\frac{1}{\frac{4}{3}u^2+1}du=\frac{\frac{4}{3}}{\frac{4}{3}} \int \frac{1}{u^2+1}du$

10. LACHEEK989 Group Title

I agree. the book (answer in the back) and wolfram alpha I believe use a substitution here of 2/sqrt3 which gets complicated..

11. Outkast3r09 Group Title

oh wait

12. Outkast3r09 Group Title

is it $\frac{1}{4/3(u^2+1)}$

13. Outkast3r09 Group Title

?

14. LACHEEK989 Group Title

1/x2+x+1 tp begin with.

15. LACHEEK989 Group Title

16. Outkast3r09 Group Title

what is problem... like that is in the book

17. Outkast3r09 Group Title

without anything done to it

18. LACHEEK989 Group Title

$\int\limits_{}^{} (x^2 -x + 2) / (x^3 -1) \ \ dx$

19. LACHEEK989 Group Title

I understand the partial fraction decomposition element to this. I am still lost at the point mentioned to complete the square. I haven't been taught yet.

20. Outkast3r09 Group Title

alright well whered the partials you used?

21. Outkast3r09 Group Title

I ask this because if you use partials you should've gotten a natural log for one of the fractions

22. wio Group Title

When you have $$u^2+1$$ it's a sign to try the trig sub $$u=\tan\theta$$.

23. LACHEEK989 Group Title

yeah the answer so far is .. $2/3 \ln (x+1) + 1/6 \ln (x^2 + x + 1) -9/2\int\limits_{}^{} dx / (x^2 + x + 1)$

24. LACHEEK989 Group Title

u = 4/sqrt3 ?

25. LACHEEK989 Group Title

sorry 2/sqrt3 = u for the substitution?

26. wio Group Title

What is the original integral?

27. LACHEEK989 Group Title

∫(x2−x+2)/(x3−1) dx

28. Outkast3r09 Group Title

the last part you have to

29. Outkast3r09 Group Title

complete the square and use trig sub

30. Outkast3r09 Group Title

$x^2+x+1$ half of b = 1/2.. $\frac{b}{4}^2=(\frac{b}{2})^2=\frac{1}{4}$

31. Outkast3r09 Group Title

if you add that you must subtract $\frac{1}{x^2+x+\frac{1}{4}+1-\frac{1}{4}}$

32. Outkast3r09 Group Title

this is where he ot $\frac{1}{(x+\frac{1}{2})^2+\frac{3}{4}}$

33. LACHEEK989 Group Title

then factor out the 3/4 in the bottom.

34. Outkast3r09 Group Title

ok now you get to where you were before right?

35. LACHEEK989 Group Title

right

36. JenniferSmart1 Group Title

:'(

37. Outkast3r09 Group Title

now let $x=\frac{a}{b}tan(\theta)$

38. LACHEEK989 Group Title

is a/b 3/4 or does should I do a substitution like 2/sqrt 3 ?