Partial Fraction Integration

[rhombuster]

I have to integrate x / (x^2+6x+13) from 4 to 8.

I'm very confused on where to start. I'm sure I need to solve the square to get (x+3)^2+4 in the denom but then I don't know how to separate the fraction.

I've found a solution but I don't even know how they did the first step. They skip right to a part where they turn it in to:
1/2(2x+6) / (x^2+6x+13)dx - 3 * dx / ((x+3)^2+4)

They then go on to do substitution in order to integrate the rest but I just don't understand that first step.

 

[Deleted member 4993]

I'm really stumped by this question. The ones before this have all been pretty straightforward. I can't seem to grasp this:

x / (x^2+6x+13)

I can see that I can complete the square on the bottom and I can make it (x+3)^2+4. After this I'm completely lost. There is a solution and the first step they take is to convert it to:

1/2(2x+6) / (x^2+6x+13) - 3 / ((x+3)^2+4)

How did they seperate it like this? Did they use Ax+B and Cx+D somehow? I can see that once that point is reached you can use substitution to figure out the rest. However, it seems they skipped the parts I don't understand in the solution.

The solution started with the fact that:

The denominator has a second degree polynomial - and its derivative is a first degree polynomial. The numerator is a first degree polynomial.

So

$\displaystyle \frac{d}{dx}[x^2 + 6x + 13] \ = \ 2x + 6$

Then

the numerator = x = ½ * (2x + 6) - 3

Then

$\displaystyle \displaystyle{\frac{x}{x^2 + 6x - 3} \ = \ \frac{\frac{1}{2}(2x+6) - 3}{x^2 + 6x - 3} }$

Now continue.....

By the way, this method is NOT the methods of partial fractions.