Integration
-----------

The `a i' (`calc-integral') [`integ'] command computes the indefinite
integral of the expression on the top of the stack with respect to a
variable.  The integrator is not guaranteed to work for all integrable
functions, but it is able to integrate several large classes of
formulas.  In particular, any polynomial or rational function (a
polynomial divided by a polynomial) is acceptable.  (Rational
functions don't have to be in explicit quotient form, however;
`x/(1+x^-2)' is not strictly a quotient of polynomials, but it is
equivalent to `x^3/(x^2+1)', which is.)  Also, square roots of terms
involving `x' and `x^2' may appear in rational functions being
integrated.  Finally, rational functions involving trigonometric or
hyperbolic functions can be integrated.

If you use the `integ' function directly in an algebraic formula, you
can also write `integ(f,x,v)' which expresses the resulting indefinite
integral in terms of variable `v' instead of `x'.  With four
arguments, `integ(f(x),x,a,b)' represents a definite integral from `a'
to `b'.

Please note that the current implementation of Calc's integrator
sometimes produces results that are significantly more complex than
they need to be.  For example, the integral Calc finds for
`1/(x+sqrt(x^2+1))' is several times more complicated than the answer
Mathematica returns for the same input, although the two forms are
numerically equivalent.  Also, any indefinite integral should be
considered to have an arbitrary constant of integration added to it,
although Calc does not write an explicit constant of integration in
its result.  For example, Calc's solution for `1/(1+tan(x))' differs
from the solution given in the *CRC Math Tables* by a constant factor
of `pi i / 2', due to a different choice of constant of integration.

The Calculator remembers all the integrals it has done.  If conditions
change in a way that would invalidate the old integrals, say, a switch
from degrees to radians mode, then they will be thrown out.  If you
suspect this is not happening when it should, use the
`calc-flush-caches' command; See Caches.

Calc normally will pursue integration by substitution or integration
by parts up to 3 nested times before abandoning an approach as
fruitless.  If the integrator is taking too long, you can lower this
limit by storing a number (like 2) in the variable `IntegLimit'.  (The
`s I' command is a convenient way to edit `IntegLimit'.)  If this
variable has no stored value or does not contain a nonnegative
integer, a limit of 3 is used.  The lower this limit is, the greater
the chance that Calc will be unable to integrate a function it could
otherwise handle.  Raising this limit allows the Calculator to solve
more integrals, though the time it takes may grow exponentially.  You
can monitor the integrator's actions by creating an Emacs buffer
called `*Trace*'.  If such a buffer exists, the `a i' command will
write a log of its actions there.

If you want to manipulate integrals in a purely symbolic way, you can
set the integration nesting limit to 0 to prevent all but fast
table-lookup solutions of integrals.  You might then wish to define
rewrite rules for integration by parts, various kinds of
substitutions, and so on.  See Rewrite Rules.