Should the Jacobian ρ in cylindrical coordinates integrate with dρ or dθ?

[zezima1]

I recently did an integral of the form:

∫∫1/ρ dρρdθ

the extra ρ between dρ and dθ is the cost of switching to cylindrical coordinates. Now I want to know, do you carry out the integration in ρ, keeping the ρ outside the integration (since it's technically a scaling factor that belongs to the integral in θ) or should I let it go under the integral to cancel out the ρ in the denominator?
Obviously two different results are achieved, and I want to know which one you must do and why...

 

[cjc0117]

You have to include it in the first integration, since the first integration is with respect to ρ. It cannot be treated as a constant in the first integration.