How Do I Decompose This Fraction in My ODE?

[smallzilla]

Hello I am stuck on an ODE involving substitution. I have done the correct substitutions, but have become stuck on decomposing the fraction.
i have the following

∫(1/x)dx + ∫(u+1)/(u^2+1)du = 0

Im stuck on breaking the u down into a partial decomposition. Could anyone offer some advice on how to start decomposing this bad boy?

Thanks

 

[ChowPuppy]

You don't need to decompose that with partial fractions.
Split up the (u+1)/(u^2+1) as u/(u^2 + 1) + 1/(u^2+1).
Now you can do a substitution on the 1st term and the 2nd term is just arctan(u).
I'm not even sure that you could do a partial fraction decomposition on that because you can't factor the denominator.

 

[smallzilla]

I see, thanks for pointing that out! I'm do for an algebra review it seems :)