Reference Guide

3-288 Full Command and Function Reference
∫ (Integrate)
Type: Function
Description: Integral Function: Integrates an integrand from lower limit to upper limit with respect to a specified
variable of integration.
The algebraic syntax for ∫ parallels its stack syntax:
∫ (lower limit, upper limit, integrand, name)
where lower limit, upper limit, and integrand can be real or complex numbers, unit objects, names, or
algebraic expressions.
Evaluating ∫ in Symbolic Results mode (flag –3 clear) returns a symbolic result. Some functions
that the calculator can integrate include the following:
• All built-in functions whose antiderivatives can be expressed in terms of other built-in functions
— for example, SIN can be integrated since its antiderivative, COS, is a built-in function. The
arguments for these functions must be linear.
• Sums, differences, and negations of built-in functions whose antiderivatives can be expressed in
terms of other built-in functions — for example, 'SIN(X)–COS(X)'.
• Derivatives of all built-in functions — for example, 'INV(1+X^2)' can be integrated because it
is the derivative of the built-in function ATAN.
• Polynomials whose base term is linear — for example, 'X^3+X^2–2*X+6' can be integrated
since X is a linear term. '(X^2–6)^3+(X^2–6)^2' cannot be integrated since X^2–6 is not linear.
• Selected patterns composed of functions whose antiderivatives can be expressed in terms of
other built-in functions — for example, '1/(COS(X)*SIN(X))' returns 'LN(TAN(X))'.
If the result of the integration is an expression with no integral sign in the result, the symbolic
integration was successful. If, however, the result still contains an integral sign, try rearranging the
expression and evaluating again, or estimate the answer using numerical integration.
Evaluating ∫ in Numerical Results mode (flag –3 set) returns a numerical approximation, and stores
the error of integration in variable IERR. ∫ consults the number format setting to determine how
accurately to compute the result.
Access: …Á (Á is the right-shift of the Ukey).
Flags: Numerical Result (–3), Number Format (–45 to –50)
Input/Output:
L4/A1 L3/A2 L2/A3 L1/A4 L1/I1
lower limit upper limit integrand 'name'
→
'symb
integral
'
L = Level; A = Argument; I = Item
Example: In Symbolic Results mode (flag –3 clear) this command sequence:
1 2 '10*X' 'X' „
returns 15.
In Numeric Results mode (flag –3 set) the above command sequence returns the numeric
approximation 15.. In addition, the variable IERR is created, and contains the error of integration
.00000000015.
See also: TAYLR, ∂, Σ
? (Undefined)
Type: Function
Description: The “undefined” symbol. Used to signify a numeric result that is not defined by the rules of
arithmetic, such as the result of dividing zero by zero, or infinity by infinity. Mathematical
operations on ? return ? as a result. Can be used in programs to check for an earlier undefined
operation.
This use of ? is unrelated to the use of ? as a spare unit in the units system. The unit ? can be used
to create new units based on it, units that can not be expressed in terms of other base units. For