Answer
See example below.
Work Step by Step
Let $f(x)=e^{x},\quad g(x)=1$
$\displaystyle \int f(x)dx=\Leftarrow e^{x}+C$
$\displaystyle \int g(x)dx=x+D$
$\displaystyle \frac{\int f(x)dx}{\int g(x)dx}=\frac{e^{x}+C}{x+D}$
has a linear term in the denominator.
But, $\displaystyle \int\frac{f(x)}{g(x)}dx=\int e^{x}dx=e^{x}+E$,
which does not have a linear term in the denominator.
So, in general,
$\displaystyle \int\frac{f(x)}{g(x)}dx\neq\frac{\int f(x)dx}{\int g(x)dx}$
