# Calc II Integration and Completing the Square

1. Sep 22, 2008

### lelandsthename

1. The problem statement, all variables and given/known data
$$\int$$$$\frac{-\frac{1}{3}x+\frac{2}{3}}{x^{2}-x+1}$$ dx

2. Relevant equations
Completing the square, partial fractions

3. The attempt at a solution
I think I need to complete the square to do this but I can't figure out how to do it. Also, do I need to separate the numerator in doing this?

This is the result of partial fractions so it is one of the last steps in my problem but I cannot understand how to do it. Any help would be fantastic!!

2. Sep 22, 2008

Complete the square on the denominator, then think about how you could factor the numerator to resemble a portion of the denominator.
To get more than this you'll need to post some work.

3. Sep 22, 2008

### jackiefrost

Hi statdad. Did you try it? Has an interesting result, huh?

jf

4. Sep 22, 2008

### lelandsthename

ok so I got the completing the square but how on earth can I continue? I just don't see it...

$$\int$$$$\frac{x-2}{(x-\frac{1}{2})^{2}+\frac{3}{4}}$$ dx

5. Sep 23, 2008

### physicsnoob93

I think if you split it up using partial fractions, it would be better.

6. Sep 23, 2008

### HallsofIvy

Staff Emeritus
Not "partial fractions" because the denominator does not factor but physicsnoob93 may mean just
$$\frac{x-2}{(x-\frac{1}{2})^2+ \frac{3}{4}}= \frac{x}{(x-\frac{1}{2})^2+ \frac{3}{4}}- \frac{2}{(x-\frac{1}{2})^2+ \frac{3}{4}}$$
The first integral requires a fairly simple substitution and the second an arctangent.

7. Sep 23, 2008

### lelandsthename

Well, I see how splitting it up makes more sense than tackling it, but I don't know what to substitute u for to get rid of both the (x - (1/2) and x. And for the arctangent, how do I go about that? I do know how to set up a trig substitution with a radical, when I must draw a triangle and find sec^2, but I am unsure in this context.