Does formula $\int_a^b \frac{f(x)}{g(x)}dx = \frac{\int_a^b {f(x)}dx}{\int_a^b{g(x)}dx}$ hold?

Question

Is the property $$\int_a^b \frac{f(x)}{g(x)}dx = \frac{\int_a^b {f(x)}dx}{\int_a^b{g(x)}dx}$$

true for all integrable $f$ and integrable and bounded $g > 0$? Intuitively I can say it shouldn't, but I couldn't find a counterexample.

Any help is appreciated!

Answer 1

If $f(x)=g(x)$ then left side is $b-a$ and right is $1$. So...