Answer
See example below.
Work Step by Step
Let $f(x)=4,\quad g(x)=e^{x}$
$\displaystyle \int f(x)dx=4x+C$
$\displaystyle \int g(x)dx=e^{x}+D$
$\displaystyle \left(\int f(x)dx\right)\left(\int g(x)dx\right)=(4x+C)(e^{x}+D)$
$=4xe^{x}+4Dx+Ce^{x}+CD\qquad (*)$
On the other hand
$\displaystyle \int[f(x)\cdot g(x)]dx=\int 4e^{x}dx=4e^{x}+E\qquad (**)$
Whatever values we assign to C,D and E, the expression (**) does not contain $4xe^{x}$, so it can not be equal to (*). This means that in general
$\displaystyle \int[f(x)\cdot g(x)]dx\neq\left(\int f(x)dx\right)\left(\int g(x)dx\right)$
