If you use the substitution $w=2 x$ to transform the integral $\int \ln (2 x) d x$ you obtain...
A. $\frac{1}{2} \int \ln (2 w) d w$
B. $2 \int \ln (2 w) d w$
c. $\int \ln (w) d w$
D. $\frac{1}{2} \int \ln (w) d w$
E. $2 \int \ln (w) d w$

Step 1 ：The question is asking to transform the integral \(\int \ln (2 x) d x\) using the substitution \(w=2 x\). This involves changing the variable of integration from \(x\) to \(w\) and adjusting the integral accordingly.

Step 2 ：The substituted integral is \(\frac{w \ln(w)}{2} - \frac{w}{2}\), which is not exactly in the form of any of the options given. However, we can simplify this expression further by factoring out \(\frac{w}{2}\).

Step 3 ：The simplified substituted integral is \(\frac{w (\ln(w) - 1)}{2}\), which is still not exactly in the form of any of the options given. However, we can see that the integral part of this expression is \(\frac{1}{2} \int \ln (w) d w\), which matches option D. The constant term \(- \frac{w}{2}\) does not affect the integral, so we can ignore it for the purposes of this question.

Step 4 ：Final Answer: \(\boxed{D. \frac{1}{2} \int \ln (w) d w}\)