Does $\frac{\int^a_b f(x)dx}{\int^a_b g(x)dx}= \int^a_b \frac{f(x)}{g(x)}dx$ for $g(x)\neq 0$ with $x\in [a,b]$

Question

Does $$\frac{\int^a_b f(x)dx}{\int^a_b g(x)dx}= \int^a_b \frac{f(x)}{g(x)}dx$$ is $$g(x)\neq 0$$ for $x\in [a,b]$

I came up with this question when I am learning Mellin's transform. I am not sure if this of my intuition is right or wrong. Please give me some guidance, many thanks!

Answer 1

There are plenty of counter examples. Here is one:

Suppose $f(x) = \frac{1}{x}$ and $g(x) = 1$ and the interval of interest is $[1,3]$. Then

$$ \int_{1}^{3} f(x) \, dx = \ln(3), \quad \int_{1}^{3} g(x) \, dx = 2, \quad \int_{1}^{3} \frac{f(x)}{g(x)} \, dx = \ln(3) $$