Primitives

Overview

A function $P$ is called a primitive (or antiderivative) of a function $f$ on an open interval $I$ if the derivative of $P$ is $f$ . That is, if $P^{'} (x) = f (x)$ for all $x \in I$ .

Leibniz preferred denoting a general primitive of $f$ as follows:

\int f (x) d x .

Note the above is conceptually distinct from an indefinite integral though the term is also often used to refer to a primitive.

Integration Techniques

Any systematic method for finding a primitive is called a technique of integration. This term is somewhat justified by the fact that the fundamental theorem of calculus reduces the problem of integrating to that of finding primitives.

By Substitution

The method of integration by substitution works by "reversing" the operations of the chain rule. Let $g : [a, b] \to I$ be continuously differentiable and $f : I \to R$ be continuous. Then:

\begin{aligned} \int f (g (x)) \cdot g^{'} (x) d x & = \int f (u) d u \\ \int_{a}^{b} f [g (x)] g^{'} (x) d x & = \int_{g (a)}^{g (b)} f (u) d u \end{aligned}

The substitutions $u = g (x)$ and $d u = g^{'} (x) d x$ are typically performed.

By Parts

The method of integration by parts works by "reversing" the operations of the product rule. Let $f$ and $g$ be both continuously differentiable. Then:

\begin{aligned} \int f (x) g^{'} (x) d x & = f (x) g (x) - \int f^{'} (x) g (x) d x \\ \int_{a}^{b} f (x) g^{'} (x) d x & = f (x) g (x) |_{a}^{b} - \int_{a}^{b} f^{'} (x) g (x) d x \end{aligned}

The substitutions $u = f (x)$ , $d u = f^{'} (x) d x$ , $v = g (x)$ , and $d v = g^{'} (x) d x$ are typically performed.

By Partial Fractions

Every proper rational function can be expressed as a finite sum of fractions of the forms

\frac{A}{(x + a)^{m}} and \frac{B x + C}{(x^{2} + b x + c)^{n}}

where $m$ and $n$ are positive integers and $A, B, C, a, b, c$ are constants with discriminant $b^{2} - 4 c < 0$ . When a rational function has been so expressed, we say it has been decomposed into partial fractions.