Integrating 1/2x: Comparing Methods and Solutions

[johann1301]

Homework Statement

∫(1/2x)dx

The Attempt at a Solution

I factor out (1/2) and i get ∫(1/2x)dx=(1/2)ln|x|+C
∫(1/2x)dx
(1/2)∫(1/x)dx
(1/2)ln|x|+C

But can't i say that u=2x and dx=du/2 and get ∫(1/2x)dx= (1/2)ln|2x| + C
∫(1/2x)dx
u=2x
∫(1/u)(du/2)
∫(1/2u)du
(1/2)∫(1/u)dx
(1/2)ln|u| + C
(1/2)ln|2x| + C

if i can do this, i get two different answers...

can i use the last method?

 

[Ibix]

ln 2x = ln x + ln 2. One of your C constants is equal to the other plus ln 2.

Both your answers are right...