Why doesn't this operation work?

Question

In my maths class we are learning about indefinite integrals, this is the problem we were working on:

$$ \int \frac{1}{2x+1}dx $$

Using u-substitution we obtain:

$$ \frac{1}{2}\ln\left | 2x+1 \right | + C $$

But why does it not work to pull out a $\frac{1}{2}$ so that we don't have to do u-substitution

$$ \frac{1}{2}\int \frac{1}{x+\frac{1}{2}}dx $$

This yields a completely different result $$ \frac{1}{2}\ln \left |x+\frac{1}{2}\right | + C_1 \neq \frac{1}{2}\ln\left | 2x+1 \right | + C_2 $$

Pulling out the $\frac{1}{2}$ seems like a completely valid move, so why does it get a completely different result?

Answer 1

They are the same because $$\ln|2x+1|=\ln|2(x+1/2)|=\ln(2)+\ln|x+1/2|$$ and the constant $\ln(2)$ goes with the constant $C$.