Calc II Integration and Completing the Square

[lelandsthename]

Homework Statement

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

Homework Equations

Completing the square, partial fractions

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!

 

[statdad]

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.

 

[jackiefrost]

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.

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

jf

 

[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

 

[physicsnoob93]

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

 

[HallsofIvy]

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.

 

[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.