Let $f\colon[a,b]\to \mathbf R$ be a continuous function, let $c\in[a,b]$ be given and consider the integral function $F$ defined on $[a,b]$ as $$ F(x)= \int_c^x f(t)\, dt. $$

Then $F$ is an antiderivative of $f$ that is, $F$ is differentiable in $[a,b]$ and $$ F'(x)=f(x)\qquad \forall x\in [a,b]. $$

The previous relation rewritten as $$ \frac{d}{dx} \int_c^x f(t)\, dt = f(x) $$ shows that the differentiation operator $\frac{d}{dx}$ is the inverse of the integration operator $\int_c^x$ . This formula is sometimes called Newton-Leibniz formula.

On the other hand if $f\colon[a,b]\to \mathbf R$ is a continuous function and $G\colon[a,b]\to \mathbf R$ is any antiderivative of $f$ , i.e. $G'(x)=f(x)$ for all $x\in[a,b]$ , then \begin{equation}\label{eq:barrow} \int_a^b f(t) \, dt = G(b)-G(a). \end{equation} This shows that up to a constant, the integration operator is the inverse of the derivative operator: $$ \int_a^x D G = G - G(a). $$

Notes