Newton-Leibnitz Integral

Inverse operation pairs. We already work comfortably with pairs of operations that undo each other: addition/subtraction $(+,-)$, multiplication/division $(\times,\div)$, and raising to a power / taking a root $\big((\ )^n, \sqrt[n]{\ }\big)$. Differentiation and integration $(d,\int)$ form exactly such an inverse pair. Integral calculus itself splits into two branches — indefinite integrals (studied in this chapter) and definite integrals (studied later) — and here we build the indefinite side: the process of obtaining a function from its derivative, called antidifferentiation.

Worked illustration (the non-uniqueness of an antiderivative). If $F(x) = x^2+5$, then $F'(x)=2x$, so $F$ is an antiderivative of $f(x)=2x$. But so are $P(x)=x^2+0$, $Q(x)=x^2+2$, and $H(x)=x^2-1$ — every one of them differentiates to the same $f(x)=2x$, because the derivative of any added constant is zero. So an antiderivative of a given $f(x)$ is never a single function — it is an entire family of infinitely many functions, each differing from the others by a constant.

Vocabulary that goes with the integral sign.

• $f(x)$, the function being integrated, is called the integrand.
• The variable $x$ appearing in $dx$ is the variable of integration (or integrator).
• The overall process of finding $\int f(x)\,dx$ is called integration, or antidifferentiation, or the Newton–Leibnitz integral.
• The integral sign $\int$ is, historically, an elongated letter S (much like $\Sigma$) — chosen precisely because it stands for a sum; this hints at the deeper (definite-integral) meaning of integration as a limit of sums, developed in later study. …