if there is an indefinite integral equality does it mean that there is an integrand equality? $$\int f(x)\,dx = \int h(x)\,dx \quad \overset{?}{\Longrightarrow} \quad f(x) = h(x)$$
I know that $$\int f(x)\,dx = \int h(x)\,dx \quad \Longrightarrow \quad \int_A f(x)\,dx = \int_A h(x)\,dx \quad $$ for any arbitrary $A$ and if $f(x),h(x)$ are continuous, then $f(x) = h(x)$, but how about more general function, discontinuous function?
Well, write the indefinite integral equality as
$$\int f(x)\,dx=F(x)=\color{blue}{\int_a^x f(t)\,dt=\int_b^x h(t)\,dt}=H(x)=\int h(x)\,dx.$$ Then by the Fundamental Theorem of Calculus, $$ {d\over dx}\int_a^x f(t)\,dt=f(x)={d\over dx}\int_b^x h(t)\,dt=h(t). $$