Integration
Integration is the reverse process of differentiation, that is, an integral
of a function f with respect to a variable x is any function g such that
D(g,x) is equal to f. The package
FunctionSpaceIntegration
provides the top-level integration operation
integrate, for integrating real-valued
elementary functions.
Unfortunately, antiderivatives of most functions cannot be expressed in
terms of elementary functions.
Given an elementary function to integrate, Axiom returns a formal integral
as above only when it can prove that the integral is not elementary and
not when it cannot determine the integral. In this rare case it prints a
message that it cannot determine if an elementary integral exists. Similar
functions may have antiderivatives that look quite different because the
form of the antiderivative depends on the sign of a constant that appears
in the function.
If the integrand contains parameters, then there may be several possible
antiderivatives, depending on the signs of expressions of the parameters.
In this case Axiom returns a list of answers that cover all possible cases.
Here you use the answer involving the square root of a when a>0 and the
answer involving the square root of -a when a<0.
If the parameters and the variables of integration can be complex numbers
rather than real, then the notion of sign is not defined. In this case all
the possible answers can be expressed as one complex function. To get that
function, rather than a list of real functions, use
complexIntegrate, which is provided
by the package
FunctionSpaceComplexIntegration.
This operation is used for integrating complex-valued elementary functions.
As with the real case, antiderivatives for most complex-valued functions
cannot be expressed in terms of elementary functions.
Sometimes integrate can involve
symbolic algebraic numbers such as those returned by
rootOf. To see how to work with these
strange generated symbols (such as %%a0), see
Using All Roots of a Polynomial.
Definite integration is the process of computing the area between the x-axis
and the curve of a function f(x). The fundamental theorem of calculus
states that if f is continuous on an interval a..b and such that D(g,x) is
equal to f, then the definite integral of f for x in the interval a..b is
equal to g(b)-g(a).
The package
RationalFunctionDefiniteIntegration
provides the top-level definite integration operation,
integrate,
for integrating real-valued rational functions.
Axiom checks beforehand that the function you are integrating is defined on
the interval a..b, and prints an error message if it finds that this is not
the case, as in the following example:
integrate(1/(x^2-2),x=1..2)
Error detected within library code:
Pole in path of integration
When parameters are present in the function, the function may or may not be
defined on the interval of integration.
If this is the case, Axiom issues a warning that a pole might lie in the
path of integration, and does not compute the integral.
If you know that you are using values of the parameter for which the
function has no pole in the interval of integration, use the string
"noPole" as a third argument to integrate.
The value here is, of course, incorrect if sqrt(a) is between 1 and 2.