Let $U(p,q)$ be $C^1(\mathbb{R^2})$ and $\frac{\partial}{\partial q}U$ the partial derivative of $U$ with respect to $q$. I know that $$\int \frac{\partial}{\partial q}U \,dq=U(p,q)+C(p)$$ where $C$ is an arbitrary function of $p$. But if I have to compute a definite integral such $$\int_{0}^{q} \frac{\partial}{\partial s}U(p,s) \,ds$$ then the answer would be $U(p,q)-U(p,0)$?
My doubt comes from the statement of my teacher that if $\frac{\partial}{\partial q}U=e^{\frac {p-q^2}{2}}$ then $$U(p,q)=e^{\frac{p}{2}}\int_{0}^{q}e^{\frac {-s^2}{2}}ds+H(p)$$ where $H$ is an arbitrary function of p.
I'm just trying to solve the pde $\frac{\partial}{\partial q}U=e^{\frac {p-q^2}{2}}$ analytically.