Definite integral: putting value before integration

Question

Let $f(x) = \int_{-c}^{c} g(k,x)dk$ and one needs to find $f(a)$. Is it allowed to substitute $x=a$ before carrying out the integration i.e. $f(a) = \int_{-c}^{c} g(k,a)dk$ is always true or not? If not, is there any special case?

Answer 1

It is allowed since you are not integrating $g$ with respect to $x$; you are treating it as a constant, and putting $x=a$ still makes it a constant.