Of course, I voted Todd Trimble's answer, but I want to address an other aspect of the question. 

Somehow, integration is *not* the inverse of differentiation. If $f:(a,b)\mapsto\mathbb R$ is differentiable, its derivative at each point $x$ is well-understandable, but $f'$ might be ugly as a function. A. Denjoy characterized (or tried to characterize ?) the functions $g:(a,b)\rightarrow\mathbb R$ that are derivatives of everywhere differentiable functions $f$. He used transfinite induction to resconstruct $f$ from $f'$. Of course, integration provides the answer when $g$ is continuous, or integrable, but not in general.