Indefinite Integrals

The notation $ \int f(x) dx = F(x)$ means that $ F'(x) = f(x)$ on some (usually specified) domain of definition of $ f(x)$.

Definition 2.2.1 (Anti-derivative)   We call $ F(x)$ an anti-derivative of $ f(x)$.

Proposition 2.2.2   Suppose $ f$ is a continuous function on an interval $ (a,b)$. Then any two antiderivatives differ by a constant.

Proof. If $ F_1(x)$ and $ F_2(x)$ are both antiderivatives of a function $ f(x)$, then

$\displaystyle (F_1(x) - F_2(x))' = F_1'(x) - F_2'(x) = f(x) - f(x) = 0.

Thus $ F_1(x) - F_2(x) = c$ from some constant $ c$ (since only constant functions have slope 0 everywhere). Thus $ F_1(x) = F_2(x) + c$ as claimed. $ \qedsymbol$

We thus often write

$\displaystyle \int f(x) dx = F(x) + c,

where $ c$ is an (unspecified fixed) constant.

Note that the proposition need not be true if $ f$ is not defined on a whole interval. For example, $ f(x) = 1/x$ is not defined at 0. For any pair of constants $ c_1$, $ c_2$, the function

$\displaystyle F(x) = \begin{cases}
\ln(\vert x\vert) + c_1 & x < 0,\\
\ln(x) + c_2 & x > 0,

satisfies $ F'(x) = f(x)$ for all $ x\neq 0$. We often still just write $ \int 1/x = \ln(\vert x\vert)+c$ anyways, meaning that this formula is supposed to hold only on one of the intervals on which $ 1/x$ is defined (e.g., on $ (-\infty,0)$ or $ (0,\infty)$).

We pause to emphasize the notation difference between definite and indefinite integration.

$\displaystyle \int_{a}^{b} f(x) dx$ $\displaystyle   =$    a specific number    
$\displaystyle \int f(x) dx$ $\displaystyle   =$    a (family of) functions    

One of the main goals of this course is to help you to get really good at computing $ \int f(x)dx$ for various functions $ f(x)$. It is useful to memorize a table of examples (see, e.g., page 406 of Stewart), since often the trick to integration is to relate a given integral to a known one. Integration is like solving a puzzle or playing a game, and often you win by moving into a position where you know how to defeat your opponent, e.g., relating your integral to integrals that you already know how to do. If you know how to do a basic collection of integrals, it will be easier for you to see how to get to a known integral from an unknown one.

Whenever you successfully compute $ F(x) = \int f(x) dx$, then you've constructed a mathematical gadget that allows you to very quickly compute $ \int_a^b f(x) dx$ for any $ a,b$ (in the interval of definition of $ f(x)$). The gadget is $ F(b) - F(a)$. This is really powerful.

William Stein 2006-03-15