The antiderivative, inverse of the derivative

To compute the definite integral, the brilliant idea is: suppose we have found a function $F(x)$ whose derivative is $f(x)$, that is $F'(x) = f(x)$. In that case the differential of $F$ is $dF = F'(x)\,dx = f(x)\,dx.$ But $f(x)\,dx$ is exactly the “little piece” we are summing in the integral! Hence:

$\int_a^b f(x)\,dx = \sum_{x=a}^{x=b} dF = F(b) - F(a).$

The sum of all the little pieces $dF$ from $a$ to $b$ is nothing but the total increase of $F$: that is, the final value $F(b)$ minus the initial value $F(a)$. It is this link — known as the fundamental theorem of calculus — that makes the integral computable.

Definition — Antiderivative

$F(x)$ is called an antiderivative of $f(x)$ if $F'(x) = f(x)$. One writes: $\mathcal{P}(f) = F \iff F' = f.$ The antiderivative is the inverse operation of the derivative: if differentiating $F$ gives $f$, finding the antiderivative of $f$ gives $F$.

Links

Topics: Integral
Concepts: Differential · Definite integral · Antiderivative · Fundamental theorem of calculus
Skills: Integrating