When is one allowed to merge "outside of integral" stuff inside integrals?

Question

When is one allowed to merge "outside of integral" stuff inside integrals?

Such as

$$\int_A f(y)dy - f(x)$$

becoming

$$\int_A [f(y)-f(x)] dy$$

Intuition says that in this case $f(x)$ doesn't depend on $y$, therefore "it doesn't alter then integral, but only by a constant". However, this is not rigorous.

Answer 1

$\int_A[f(y)-f(x)]dy=\int_A f(y)dy-\int_A f(x)dy=\int_A f(y)dy-f(x)m(A)$ wher $m$ is Lebesgue measure. If $m(A)=1$ then you write $\int_A[f(y)-f(x)]dy= \int_A f(y)dy-f(x)$