Finding the integral using partial fractions

[sdoyle]

Homework Statement

Solve the integral x/x^2+4x+13

Homework Equations

I think that you would use partial fractions but I'm not really sure. I know that you need to complete the square on the denominator.

The Attempt at a Solution

The completed square would be (x+2)^2+9. I don't know what to do now b/c of the x term on top. Help!

 

[HallsofIvy]

Since the denominator cannot be factored (in terms of real coefficients), there is no "partial fractions".

Let u= x+ 2 so x= u-2. Then the fraction becomes
$$\frac{u-2}{u^2+ 9}= \frac{u}{u^2+ 9}-2\frac{1}{u^2+ 9}$$
Let $v= u^2+ 9$ to integrate the first and the second is an arctangent.

 

[Gregg]

$\frac{x}{x^2+4x+13} = -\frac{2}{x^2+4x+13} + \frac{x+2}{x^2+4x+13}$

$\int \frac{dx}{x^2+a^2} = \frac{1}{a}ArcTan\frac{x}{a} + C$

$\int \frac{x'}{x} = Ln|x| + C$

That's all you need.

Edit: Method below a lot better.