calc: Integration
11.5.2 Integration
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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; 
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. Rewrite Rules.
Info Catalog
calc: Differentiation
calc: Calculus
calc: Customizing the Integrator