Integral

An integral comes in two flavours. The definite integral of $f$ from $a$ to $b$,

$\int_a^b f(x)\,dx,$

equals the (signed) area between the curve $y = f(x)$ and the x-axis on $[a, b]$. The indefinite integral $\int f(x)\,dx$ is the family of antiderivatives — functions whose derivative is $f$.

The two are linked by the Fundamental Theorem of Calculus: if $F$ is any antiderivative of $f$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

Integration techniques (substitution, integration by parts, partial fractions, trigonometric substitution) form the bulk of a first calculus course. Most "real-world" antiderivatives cannot be expressed in elementary functions and require numerical methods.