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10.5.2 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
prompted-for 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.

With an argument (C-u a i), this command will compute the definite
integral of the expression on top of the stack. In this case, the
command will again prompt for an integration variable, then prompt
for a lower limit and an upper limit.

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.

Next: Customizing the Integrator, Previous: Differentiation, Up:
Calculus   [Contents][Index]