Answer
False
Work Step by Step
For example, let f(x) = g(x) = x
Therefore, LHS = $ \int x . x dx =\int x^2 dx = \frac{x^3}{3} $
whereas RHS = $ \int x dx \int x dx = \frac{x^2}{2}\frac{x^2}{2} = \frac{x^4}{4}$
Clearly, LHS$\ne$RHS and hence the given statement is proven false by counter-example
