There exist two functions f, g satistfying: $\int f(x)·g(x)dx = \int f(x)dx ·\int g(x) dx$?

Question

I know that the integral of two functions is not the product of the integrals, but there exist two functions f, g satistfying:

$$\int f(x)·g(x)dx = \int f(x)dx ·\int g(x) dx$$

Answer 1

Yes.

Let $$f(x)=g(x)=e^{2x}$$

$$\int f(x)g(x)dx = \int e^{4x}dx =e^{4x}/4$$

Also $$ \int f(x) dx \int g(x) dx =(e^{2x}/2)(e^{2x}/2)=e^{4x}/4.$$