Can anyone please help me with this INTEGRATION. I'm stuck o_o?

Question

Integrate: http://img.villagephotos.com/p/2006-10/1221148/1orig.JPG

I did long ÷ & completed the square for the denominator to get the integral: http://img.villagephotos.com/p/2006-10/1221148/2nd.JPG

I then used a regular u-substitution u = x + 2 & got:
http://img.villagephotos.com/p/2006-10/1221148/3rd.JPG

I did another substitution v = (u / √5) to get:
http://img.villagephotos.com/p/2006-10/1221148/4stuck.JPG

That's as far as I got. I thought I could then use the arctan formula, but I can't figure out how to manipulate the problem to get in the proper form to use arctan.
I tried trig substitution (build a triangle), but I got stuck with that too.

Did I even approach this problem the right way?

Any helpful input is greatly appreciated. Thank you for your time.

To Fatty Dave: I cannot separate the fraction because it is an addition not a multiplication in the numerator. If there were a way I could make it into a multiplication, then that would work.

Answer 1

Looks like you're on the right track.
I looked it up, you do use arctan on part of it, and ln on the rest.
Way to go!

Answer 2

From where you stopped, separate the fraction into two fractions (separate numerators). The one with the 1 on top will become arctan, and with the other one u'll have to use partial fractions again.

Answer 3

∫6x^2/(x^2+4x+9) dx
= 6[∫1 - (4x+9)/(x^2+4x+9) dx]
= 6[x - ∫(2(2x+4)+1)/(x^2+4x+9) dx
= 6[x- 2ln(x^2+4x+9) - ∫1/((x+2)^2+5)) dx]
= 6[x - 2ln(x^2+4x+9) - (1/√5)arctan[(x+2)/√5]