Let $f\in C(a,b],$$\lim_{x\to a}f(x) = \infty$ and $\int_a^bf(x)dx$ exists (finite) as Improper integral.
Can the expression $\int_a^xf(t)dt$ be considered as a Primitive of the function $f(x)$ on a half-interval $(a,b]$?
As I see, the function $F(x)=\int_a^xf(t)dt$ is defined on the segment $[a,b]$.
Due to additivity, it can be investigated as ordinary Riemann integral in each point $x\in(a,b]$ using $F(x+\delta )-F(x)=\int_x^{x+\delta}f(t)dt,$ where $[x,x+\delta]\subset(a,b]$ (for $b:$$[b-\delta,b]\subset(a,b]$).
Therefore the function $F(x)$ is continuously differentiable on $(a,b]$ and $F'(x)=f(x)$ here.
Have I made a mistake somewhere?
If not, can I relax the restrictions on the $f(x)$?