How Do I Decompose This Fraction in My ODE?

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SUMMARY

The discussion centers on decomposing the fraction (u+1)/(u^2+1) in the context of solving an ordinary differential equation (ODE). The correct approach is to split the fraction into two simpler components: u/(u^2 + 1) and 1/(u^2 + 1). The first term can be solved using substitution, while the second term directly leads to the arctangent function, arctan(u). Partial fraction decomposition is not applicable here due to the irreducibility of the denominator.

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

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