Pulling out a new variable inside an integral

Suppose I have$$f(t)\le\int_0^t\frac{(f(s)+c)}{g(s)}ds$$ where $c$ is constant.

Then, by writing $f(t)=\int_0^tf'(s)ds=\int_0^tdf(s)$$=\int_0^td(f(s)+c)$, is it possible to write the above inequality as$$\int_0^t\frac{d(f(s)+c)}{f(s)+c}\le\int_0^t\frac1{g(s)}ds$$

Assuming the conditions of fundamental theorem of calculus, continuity and non-vanishing of $f(t)+c$ are satisfied, is the above true.

I saw this procedure in a paper I was studying and was dumb founded by it. Any clarifications? Thanks beforehand.