Integration: TI-86 & Casio Algebra 2.0 Plus

[Albert Chan]

Hi, KeithB

The problem for HP71B is that integrand, after u-transformed, is still inverted U shaped.

We would really like bell shaped, for efficient integration.
One way is to apply u-transform twice. (1 explicit, 1 internal)

10 DEF FNF(X)=ATAN(ATANH(X))/X
20 DEF FNU(U)=FNF(U*U*(3-2*U))*6*U*(1-U)
30 DISP INTEGRAL(0, 1, 1E-10, FNU(IVAR))
>run
 1.02576051093

Or, remove cusp at x=1. Let x=1-u^2, turn curve close to a straight line.
Internal u-transform will again get a bell shaped curve.

>20 DEF FNU(U)=FNF(1-U*U)*2*U
>run
 1.02576051093

Or, pick a curve close to what we had, and we know its area.
Let y = atanh(x) --> x = tanh(y) --> dx = sech(y)^2 dy

∫(atan(atanh(x))/x, x = 0 .. 1) = ∫(atan(y)/tanh(y) * sech(y)^2, y = 0 .. ∞)

Cas> series(atan(y), y)      → y - 1/3*y^3 + 1/5*y^5 + y^6*order_size(y)
Cas> series(tanh(y),y)       → y - 1/3*y^3 + 2/15*y^5 + y^6*order_size(y)

atan(y)/tanh(y) = 1 + 1/15*y^4 + ... ≈ 1 if y is small,
And, we know ∫(sech(y)^2, y = 0 .. ∞) = ∫(1, x = 0 .. 1) = 1

2 curves are close, and its difference give a nice bell curve, easy to integrate.

plot (atan(y)/tanh(y)-1) * sech(y)^2, y = 0 ..​ 10

>INTEG(.01, 100, 1E-9, (ATAN(IX)/TANH(IX)-1) / COSH(IX)^2)
 2.57605109301E-2
>1+RES
 1.02576051093