Difference between integrals

Question

What's the difference between $\int_0^xf(x)dx $ and $\int_0^xf(x')dx'$? Please give examples with the explanation! Why ist wrong to integrate $\int_0^xf(x)dx $

Answer 1

$\int_0^xf(x)dx$ is a wrong expression, because you cannot integrate over $x$ on an interval $[0,x]$ (if $x$ increases, then the boundary would increase too).

$\int_0^xf(x')dx'$ is the way to write it down (or $\int_0^xf(t)dt$, just choose any dummy variable you like). Then you get $\int_0^xf(x')dx'=F(x)-F(0)$ by the fundamental theorem of calculus, where $F'(x)=f(x)$.