Numerical integration
Numeric integration of symbolic expressions is performed in newRPL via the NUMINT
command which implements the Adaptive Simpson's method.
NUMINT
accepts four arguments:
 the monovariate function to integrate, either in symbolic or program form;
 the lower integration limit;
 the upper integration limit;
 the error tolerance.
If the function to integrate is expressed in symbolic form it must respect a precise syntax:
 it must be written as an equation;
 the left side must be in the form
func(var)
wherevar
is the integrating variable andfunc
is the function's name;  the right side is a function, expressed in terms of
var
.
For example valid expressions are:
'F(X)=X*LN(X)' 'G(Z)=Z^22*COS(Z)' 'VEL(T)=ACC*T'
As shown above, the functions may refer to global or local variables; the function's name is only descriptive and bears no relevance to the calculation.
Alternatively, the function can be written as a program which accepts exactly one numeric argument and returns exactly one numeric result. The expressions above can be rewritten as:
« DUP LN * » « DUP SQ SWAP COS 2 *  » « 'ACC' RCL * »
The integration limits can be either real or complex finite numbers; symbolic constants are accepted and silently converted to numerical values.
The error tolerance is a real number used to specify the required precision of the calculation: when two successive iterations differ by a value which is less than the tolerance the calculation stops.
Angles and trigonometric expressions
When a real number is input to a trigonometric function newRPL assumes that it is an angle expressed in the current angular mode; however the trigonometric functions are meant to process quantities expressed in radians. In other words the following transformations are implicitly applied:
 in
RAD
mode $ θ=θ_{r}\, $;  in
DEG
mode $ θ=\frac{\pi}{180}{θ_°}\, $;  in
GRAD
mode $ θ=\frac{\pi}{200}{θ_g}\, $.
where $θ_{r}\,$, $θ_{°}$ and $θ_{g}$ are the quantities entered by the user and $θ$ is what is actually fed to the trigonometric functions.
From this, two important consequences derive:
 the antiderivative changes according to the angular mode. Let's consider e.g. the function $ f(θ)=\sin{θ} $ and apply the transformations above:
 in
RAD
mode $ f(θ_r) $ is equivalent to $ f(θ) $ which differentiates to $ f(θ_r)\,dθ_r=\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=\cos{θ}\, $;  in
DEG
mode $ f(θ_°) $ is actually $ f\left(\frac{\pi}{180}θ_°\right) $ which differentiates to $ f\left(\frac{\pi}{180}θ_°\right)\,dθ_°=\frac{180}{\pi}\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=\frac{180}{\pi}\cos{θ}\, $;  in
GRAD
mode $ f(θ_g) $ is actually $ f\left(\frac{\pi}{200}θ_g\right) $ which differentiates to $ f\left(\frac{\pi}{200}θ_g\right)\,dθ_g=\frac{200}{\pi}\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=\frac{200}{\pi}\cos{θ}\, $;
 no inverse transformation is applied to the resulting output: this is mathematically correct, but can be disconcerting if one is not immediately aware of the implicit variable substitution.
In conclusion, unless the user knows exactly what he/she is doing it's advisable to perform numeric integration of trigonometric expressions in RAD
mode.
Example 1: Bounded function on closed interval
pic  $$ \int^{2}_{0} x^{10}e^{\left(4x^{3}3x^{4}\right)} \,dx $$  Input:16 SETPREC 'F(X)=X^10*EXP(4*X^33*X^4)' 0 2 tol NUMINT 

tol=10^{4}  7.258 376 114 514 225 …  Δ = 1.9·10^{5}  
tol=10^{8}  7.258 395 173 115 920 …  Δ = 2.5·10^{9}  
tol=10^{12}  7.258 395 170 615 141 …  Δ = 8.5·10^{13}  
tol=10^{16}  7.258 395 170 614 323 …  Δ = 3.2·10^{14}  
Truncated at 16 digits  7.258 395 170 614 291 … 
This integral is interesting because the shape of the function in the interval is flat almost everywhere except for a narrow peak which is however captured by the algorithm.