Answer
\begin{aligned}
\int \frac{2^{\ln z^{3}} d z}{16 z} = \frac{ 1}{16 \ln 8}8^{\ln z}+c\\
\end{aligned}
Work Step by Step
Given $$\int \frac{2^{\ln z^{3}} d z}{16 z}$$
So, we get
\begin{aligned}
I&=\int \frac{2^{\ln z^{3}} d z}{16 z}\\
&=\int \frac{2^{3\ln z } d z}{16 z}\\
&=\int \frac{8^{\ln z } d z}{16 z}\\
\end{aligned}
Let $$ y=8^{\ln z } \Rightarrow dy=\frac{\ln 8}{z}8^{\ln z }dz \Rightarrow \frac{1}{z}8^{\ln z }dz=\frac{1}{\ln 8}dy $$
So, we get
\begin{aligned}
I&=\int \frac{8^{\ln z } d z}{16 z}\\
&=\int \frac{ dy}{16 \ln 8}\\
&= \frac{ 1}{16 \ln 8} \int dy\\
&= \frac{ 1}{16 \ln 8}y+c\\
&= \frac{ 1}{16 \ln 8}8^{\ln z}+c\\
\end{aligned}
