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The command RINT computes the indefinite integral of
rational functions. It takes two arguments: the expression to
integrate and the integration variable. If the expression to
integrate contains non-rational subexpressions that depend on the
integration variable or contains algebraic extensions (like
) then RINT
produces a "Bad Argument Value" error. Even
though it is not explicit in the output, like any indefinite
integral, the integral returned by RINT is defined up to
an additive constant. That is, the general solution for the
indefinite integral is the output of RINT plus an
arbitrary constant.
In general, the indefinite integral of a rational function will have a rational part and a logarithmic part. The rational part is a ``regular'' rational function in the integration variable, and the logarithmic part is a sum of logarithms whose arguments are polynomials in the integration variable and whose coefficients are constants. E.g.,
It is always possible to compute the rational part of the integral and RINT uses Horowitz's algorithm to compute it quickly without computing the partial fraction expansion of the rational function. For instance,
On the other hand, the coefficients in the logarithmic part are solutions of potentially high-degree equations and cannot always be represented analytically (in closed-form). RINT gives an analytical solution only if the coefficients are solutions of equations of degree two or less, i.e., if the coefficients can be expressed exactly in terms of rational numbers and radicals. Otherwise RINT leaves the corresponding part of the integral unsolved. For instance, RINT completely solves the following integral since the it can be given explicitely in terms of radicals and fractions,
In the contrary, RINT leaves the following integral unsolved because the solution can only be expressed in terms of the roots of an equation of degree three,
When appropriate, to avoid logarithms with complex arguments and coefficients, RINT uses arctangents in the logarithmic part of the integral. E.g.,
Because of the limited speed of the calculator, RINT does not use the general Rothstein-Trager method to compute the logarithmic part of the integral and in some cases will fail to give an analytical solution when one exists. For instance, RINT fails to solve completely the following integral,
even though the integral of the last term can be given analytically as
Note however that RINT never introduces unnecessary algebraic extensions to express the integral and can always solve integral whose logarithmic part only entails logarithms with polynomials of degree two or less, regardless of the degree of the rational function itself.
Next: Symbolic
matrix manipulation Up: Commands Previous: Partial fraction expansion