Some time ago I learned about the following property of integrals:
If $ f $ and $ g $ are bounded, integrable functions on $\color{red}{[a, b]}$, then so is $f + g$ and
$$ \displaystyle \int_\color{red}a^\color{red}b [f(x) + g(x)] dx = \displaystyle \int_\color{red}a^\color{red}b f(x) dx + \displaystyle \int_\color{red}a^\color{red}b g(x) dx $$
My question:
Can this sum property of integrals also be stated for indefinite integrals (without the extremes of integration from $\color{red}a$ to $\color{red}b$) and without specifying a domain ($\color{red}{[a, b]}$) for the functions?
In other words, would the following property be equivalent?
If $ f $ and $ g $ are bounded, integrable functions, then so is $f + g$ and
$$ \displaystyle \int [ f(x) + g(x)] dx = \displaystyle \int f(x) dx + \displaystyle \int g(x) dx $$
I came up with the following example which suggests that the second theorem works just as well, but there may be counter-examples or other details that I may have overlooked:
$$\int \left[ x^2 + e^x \right] dx = \frac{x^3}{3} + e^x + C = \int{x^2 \, dx} + \int e^x \, dx$$
The statements$$\int_a^bf(x)+g(x)\,\mathrm dx=\int_a^bf(x)\,\mathrm dx+\int_a^bg(x)\,\mathrm dx\tag1$$and$$\int f(x)+g(x)\,\mathrm dx=\int f(x)\,\mathrm dx+\int g(x)\,\mathrm dx\tag2$$mean distinct (although related) things. In fact, $(1)$ is about the value of integrals, whereas $(2)$ states that if you add an antiderivative of $f$ to an antiderivative of $g$, what you get is an antiderivative of $f+g$.