calc: Customizing the Integrator
11.5.3 Customizing the Integrator
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Calc has two built-in rewrite rules called ‘IntegRules’ and
‘IntegAfterRules’ which you can edit to define new integration methods.
Rewrite Rules. At each step of the integration process, Calc
wraps the current integrand in a call to the fictitious function
‘integtry(EXPR,VAR)’, where EXPR is the integrand and VAR is the
integration variable. If your rules rewrite this to be a plain formula
(not a call to ‘integtry’), then Calc will use this formula as the
integral of EXPR. For example, the rule ‘integtry(mysin(x),x) :=
-mycos(x)’ would define a rule to integrate a function ‘mysin’ that acts
like the sine function. Then, putting ‘4 mysin(2y+1)’ on the stack and
typing ‘a i y’ will produce the integral ‘-2 mycos(2y+1)’. Note that
Calc has automatically made various transformations on the integral to
allow it to use your rule; integral tables generally give rules for
‘mysin(a x + b)’, but you don’t need to use this much generality in your
‘IntegRules’.
As a more serious example, the expression ‘exp(x)/x’ cannot be
integrated in terms of the standard functions, so the “exponential
integral” function ‘Ei(x)’ was invented to describe it. We can get Calc
to do this integral in terms of a made-up ‘Ei’ function by adding the
rule ‘[integtry(exp(x)/x, x) := Ei(x)]’ to ‘IntegRules’. Now entering
‘exp(2x)/x’ on the stack and typing ‘a i x’ yields ‘Ei(2 x)’. This new
rule will work with Calc’s various built-in integration methods (such as
integration by substitution) to solve a variety of other problems
involving ‘Ei’: For example, now Calc will also be able to integrate
‘exp(exp(x))’ and ‘ln(ln(x))’ (to get ‘Ei(exp(x))’ and ‘x ln(ln(x)) -
Ei(ln(x))’, respectively).
Your rule may do further integration by calling ‘integ’. For
example, ‘integtry(twice(u),x) := twice(integ(u))’ allows Calc to
integrate ‘twice(sin(x))’ to get ‘twice(-cos(x))’. Note that ‘integ’
was called with only one argument. This notation is allowed only within
‘IntegRules’; it means “integrate this with respect to the same
integration variable.” If Calc is unable to integrate ‘u’, the
integration that invoked ‘IntegRules’ also fails. Thus integrating
‘twice(f(x))’ fails, returning the unevaluated integral
‘integ(twice(f(x)), x)’. It is still valid to call ‘integ’ with two or
more arguments, however; in this case, if ‘u’ is not integrable, ‘twice’
itself will still be integrated: If the above rule is changed to ‘... :=
twice(integ(u,x))’, then integrating ‘twice(f(x))’ will yield
‘twice(integ(f(x),x))’.
If a rule instead produces the formula ‘integsubst(SEXPR, SVAR)’,
either replacing the top-level ‘integtry’ call or nested anywhere inside
the expression, then Calc will apply the substitution ‘U = SEXPR(SVAR)’
to try to integrate the original EXPR. For example, the rule ‘sqrt(a)
:= integsubst(sqrt(x),x)’ says that if Calc ever finds a square root in
the integrand, it should attempt the substitution ‘u = sqrt(x)’. (This
particular rule is unnecessary because Calc always tries “obvious”
substitutions where SEXPR actually appears in the integrand.) The
variable SVAR may be the same as the VAR that appeared in the call to
‘integtry’, but it need not be.
When integrating according to an ‘integsubst’, Calc uses the equation
solver to find the inverse of SEXPR (if the integrand refers to VAR
anywhere except in subexpressions that exactly match SEXPR). It uses
the differentiator to find the derivative of SEXPR and/or its inverse
(it has two methods that use one derivative or the other). You can also
specify these items by adding extra arguments to the ‘integsubst’ your
rules construct; the general form is ‘integsubst(SEXPR, SVAR, SINV,
SPRIME)’, where SINV is the inverse of SEXPR (still written as a
function of SVAR), and SPRIME is the derivative of SEXPR with respect to
SVAR. If you don’t specify these things, and Calc is not able to work
them out on its own with the information it knows, then your
substitution rule will work only in very specific, simple cases.
Calc applies ‘IntegRules’ as if by ‘C-u 1 a r IntegRules’; in other
words, Calc stops rewriting as soon as any rule in your rule set
succeeds. (If it weren’t for this, the ‘integsubst(sqrt(x),x)’ example
above would keep on adding layers of ‘integsubst’ calls forever!)
Another set of rules, stored in ‘IntegSimpRules’, are applied every
time the integrator uses algebraic simplifications to simplify an
intermediate result. For example, putting the rule ‘twice(x) := 2 x’
into ‘IntegSimpRules’ would tell Calc to convert the ‘twice’ function
into a form it knows whenever integration is attempted.
One more way to influence the integrator is to define a function with
the ‘Z F’ command (Algebraic Definitions). Calc’s integrator
automatically expands such functions according to their defining
formulas, even if you originally asked for the function to be left
unevaluated for symbolic arguments. (Certain other Calc systems, such
as the differentiator and the equation solver, also do this.)
Sometimes Calc is able to find a solution to your integral, but it
expresses the result in a way that is unnecessarily complicated. If
this happens, you can either use ‘integsubst’ as described above to try
to hint at a more direct path to the desired result, or you can use
‘IntegAfterRules’. This is an extra rule set that runs after the main
integrator returns its result; basically, Calc does an ‘a r
IntegAfterRules’ on the result before showing it to you. (It also does
algebraic simplifications, without ‘IntegSimpRules’, after that to
further simplify the result.) For example, Calc’s integrator sometimes
produces expressions of the form ‘ln(1+x) - ln(1-x)’; the default
‘IntegAfterRules’ rewrite this into the more readable form ‘2
arctanh(x)’. Note that, unlike ‘IntegRules’, ‘IntegSimpRules’ and
‘IntegAfterRules’ are applied any number of times until no further
changes are possible. Rewriting by ‘IntegAfterRules’ occurs only after
the main integrator has finished, not at every step as for ‘IntegRules’
and ‘IntegSimpRules’.
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