1. The Indefinite Integral

Recall that if I is any interval and f is a function defined on I, then a function F on I is an antiderivative of f if F'(x) = f(x) for all x ∈ I. A corollary of the Mean Value Theorem tells us that if G is any other antiderivative of f on I, then there is a constant C such that G(x) = F(x) + C.

Definition

Let I be an interval and f a function defined on I. Then the indefinite integral of f is defined to be the set of all antiderivatives of f on I. We denote this set by:

We call f the integrand and x the variable of integration.

Since all the antiderivatives of f differ only by a constant, it is customary to write:

where F is any particular antiderivative, and C is the constant of integration which implicitly takes on all real number values.

2. The Definite Integral

Definition

Let f be a function defined on a closed interval [a, b]. Given any n we let Δx=(b-a)/n, and we let xk = a + k Δx. For each k, select sample points xk* ∈ [xk-1, xk]. We define the definite integral of f on [a, b] to be

provided this limit exists and does not depend on the choice of sample points xk*.

If the limit depends upon the choice of sample points, or is undefined for any choice of sample points, then the integral does not exist.

We call a the lower limit of the integral, and b the upper limit. We call f the integrand.

We also make the defintions:

The integral sign "∫" is an archaic "S" and stands for "sum" (as does ∑, of course).

If it exists, the definite integral gives you a number as its result. In this regard, x is a "dummy variable" and the dx reminds us what variable we are integrating with respect to. We can just as easily replace x with t and write:

and the numerical value of the definite integral is unchanged.

Geometrically, if f(x) ≥ 0 we interpret the integral as the area between the graph y=f(x), the x-axis, and the lines x=a and x=b. More generally, we think of the integral as a "signed" area, where the integral is equal to the area above the x-axis less the area below the x-axis.

The sum:

in the definition of the definite integral is called a Riemann sum; the definite integral is sometimes called a Riemann integral (to distinguish it from other more general integrals used by mathematicians). Whenever you have a limit as n goes to infinity of a Riemman sum, you can replace it by an appropriate integral. This observation is critical in applications of integration.

For certain simple functions, you can calculate an integral directly using this definition. However, in general, you will want to use the fundamental theorem of calculus and the algebraic properties of integrals.

3. Rules of Integration

Let f and g be functions and let a, b and c be constants, and assume that for each fact all the indicated definite integrals exist. Then the following are true:

  1. Constants can be pulled out of integrals:

  2. The integral of the sum of two functions equals the sum of the integrals of each function:

  3. The integral of the difference of two functions equals the difference of the integrals of each function:

  4. The integral from a to b of a function equals the integral from a to c plus the integral from c to b:

Note that there are no general rules for integrals of products and quotients. Such integrals can sometimes, but not always, be calculated using substitution or integration by parts.

We can also give some facts about inequalities involving definite integrals.

  1. If f(x) ≥ 0 on [a, b] then

  2. If f(x) ≤ g(x) on [a, b] then

  3. The absolute value of an integral is less than the integral of the absolute value.

4. Basic Integrals

The following are some basic indefinite integrals. The table can also be used to find definite integrals using the fundamental theorem of calculus. More comprehensive tables can usually be found in a calculus textbook, but the ones listed here are good ones to know without having to look up a reference.

Integrals of Powers

Integrals of Exponentials

Integrals of Trigonometric Functions

Integrals Involving Inverse Trigonometric Functions