Why can't I u-sub for the second part of this integral $\int \ln(2x+4)$? I was reviewing one of my integral problems, and I ran into this problem: 
$$\int \ln(2x+4)dx $$
I did integration by parts and got this:
$$\ln(2x+4)x - \int \frac{2x}{2x+4}du $$ 
For the second integral, I did a u-sub:
$$ \frac{1}{2} \int \frac{u-4}{u} dx $$
Where $u=2x+4$, but the issue here is that my review notes told me that I can't do a sub here. The review notes told me to do a polynomial division instead. If I do it my way, I get the answer wrong but there was no explanation of why I can't do a sub for that type of integral. 
 A: Dividing top & bottom by $2$ and fairly directly integrating your original integral gives
$$\begin{equation}\begin{aligned}
\int \frac{2x}{2x+4} dx & = \int \frac{x}{x+2} dx \\
& = \int \frac{x + 2 - 2}{x+2} dx \\
& = \int \left(1 - \frac{2}{x + 2}\right) dx \\
& = x - 2\ln|x + 2| + C_1
\end{aligned}\end{equation}\tag{1}\label{eq1A}$$
Note I effectively did polynomial division where I used that $x = x + 2 - 2$ in the numerator in the second line. Next, using your substitution of $u = 2x + 4$ gives $du = 2dx \implies dx = \frac{du}{2}$. Thus, you then get
$$\begin{equation}\begin{aligned}
\int \frac{2x}{2x+4} dx & = \frac{1}{2}\int \frac{u - 4}{u}du \\
& = \frac{1}{2}\int \left(1 - \frac{4}{u}\right)du \\
& = \frac{1}{2}\left(u - 4\ln|u|\right) + C_2 \\
& = \frac{1}{2}\left((2x + 4) - 4\ln|2x + 4|\right) + C_2 \\
& = x + 2 - 2\ln|2(x + 2)| + C_2 \\
& = x - 2(\ln|x + 2| + \ln(2)) + 2 + C_2 \\
& = x - 2\ln|x + 2| + (2\ln(2) + 2 + C_2)
\end{aligned}\end{equation}\tag{2}\label{eq2A}$$
Note you made a small mistake in your question text where you used $dx$ instead of $du$. As you can see, you get the same result, with $C_1$ from \eqref{eq1A} being equal to $2\ln(2) + 2 + C_2 = \ln(4) + 2 + C_2$ from \eqref{eq2A}, so using substitution does work.
There's no reason why you can't use substitution. Instead, as several question comments have indicated, it's fairly likely your review notes stated to use polynomial division for you to practice that technique instead.
