Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Question

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$?

My teacher asked us to give examples to prove that this is not true but I was curious as to what nontrivial functions satisfy this. Any idea how to solve?

Thanks!!

Answer 1

Let $F(t) =\int f(t)dt , G(t) =\int g(t)dt $ then the your equation becomes $$\int f(t) g(t) dt = F(t) G(t) $$ and after differntiation $$F' G' =F'G+G'F $$ and hence $$1=\frac{G}{G' } +\frac{F}{F'} $$ if you take for example $$G(t) =\ln t $$ then you obtain $$1=\frac{\ln t}{\frac{1}{t} } +\frac{F}{F'} $$ hence $$\frac{F}{F'} =1-t\ln t$$ hence $$\ln F(t) = \int \frac{1}{1-t\ln t} dt$$ thus $$F(t) =e^{\int \frac{1}{1-t\ln t} dt}.$$ So there is a lot of pairs $(f,g) $ of functions satisfying this equation.