This is a just a constant (21ln2) different from your other answer so they both differentiate to give the original function.
The constant of integration can take any form. If you add 21lnc to your first answer you get:
21lnx+21lnc=21lncx
So the second answer is just the case where c=2.
You have solved something very simple which has been troubling me fore about two years, but I never asked anyone
If I had to integrate (2x)-1 to use it as an integrating factor in a DE (so I will raise it to the integral power e) then the final answer I get as the solution to the equation will be different depending on whether I have ln 2x or ln x. As eln 2x = 2x but eln x = x so I will be pretty much end up with different answers to my solution, but there can only be one correct general solution and both methods I have used to get there are correct ?
You have solved something very simple which has been troubling me fore about two years, but I never asked anyone
If I had to integrate (2x)-1 to use it as an integrating factor in a DE (so I will raise it to the integral power e) then the final answer I get as the solution to the equation will be different depending on whether I have ln 2x or ln x. As eln 2x = 2x but eln x = x so I will be pretty much end up with different answers to my solution, but there can only be one correct general solution and both methods I have used to get there are correct ?
You multiply the whole equation by the integrating factor, so a constant multiplier won't make a difference - you can divide it back out again!