# When do integrable functions have a primitive?

Studying hyperbolic partial differential equations, we arrive, in a certain calculation, to the following doubt: every integrable function has a primitive?

If $$u_0$$ is integrable, then $$\exists v_0$$ such that $$\int_b^a v'_0 dt = \int_b^a u_0 dt$$?

I think no, but maybe there is some theorem that determines when this holds. And maybe this theorem aplies to our case.

Many thanks for any help!

• For fixed $a,b$, there is a constant $C$ such that $\int_b^a C dt = \int_b^a u_0 dt$. So perhaps you mean something more difficult. Such as: there exists $v_0$ such that your equation holds for all $a,b$? – GEdgar Apr 20 at 20:39
• @Gedgar, yes, this is what i am trying, for all a and b. Thanks. – Na'omi Apr 21 at 19:15

Something similar is correct. We cannot arrange that $$v_0$$ is differentiable everywhere. But we can arrange that is it differentiable almost everywhere.
If $$u_0$$ is integrable on the interval $$[0,1]$$, then $$v_0(x) = \int_a^x u_0(t)\;dt,\qquad 0 \le x \le 1$$ satisfies $$v_0'(x) = u_0(x)$$ for almost all $$x \in [0,1]$$. And therefore $$\int_a^b v_0'(x)\;dx = \int_a^b u_0(x)\;dx,\qquad 0 \le a \le b \le 1$$