(35S) Integration with infinite limits ∫+∞, -∞

[Roberto Volpi]

In the HP35S manual at E-3 there is a workaround attempting to solve the infamous integral "∫ from 0 to ∞ of the expression xe^-x", but the proposed solution to change ∞ with 10E3 is far from ideal.

This short program is inspired by "Tips and Programs for the HP-32S"" published by Synthetix in 1988.

This program turns the original integral:"∫ from a to b of f(x) dx"
Into a new: "∫ from ATANa to ATANb of f(TANθ)/COS^2(θ) dθ"
so you can enter π/2 instead of +∞ and -π/2 instead of -∞.

It assumes that LBL F is dedicated to the function to be integrated.


LINE INSTRUCTION

I001 LBL I
I002 (MODE) RAD
I003 RCL X
I004 ENTER
I005 COS
I006 X^2
I007 X=0?
I008 RTN
I009 STO A
I010 X<>Y
I011 TAN
I012 STO X
I013 XEQ F001
I014 RCL A
I015 /
I016 RTN

INSTRUCTIONS:

1. Enter the function in LBL F (as EQN or in RPN. Please take into account that EQN mode is more visual appealing, but RPN mode is approx. 30% faster.

2. Input lower limit in RAD, press ENTER, and input upper limit in RAD.

3. Press FN= I ∫ , chose x to integrate, and wait patiently for the answer.

NOW YOU CAN EASILY SOLVE THE INFAMOUS INTEGRATION WITH THE INFINITE LIMIT "∫ from 0 to ∞ of the expression xe^-x":

1. Enter in LBL F the expression as: LBL F, RCL X, ENTER, +/-, e^x, *, RTN.

2. Enter 0 as lower limit, and press π, 2, / as upper limit.

3. Press FN= I ∫ , chose x to integrate, and you obtain in less than 3 minutes the right answer "1".

If you need to integrate from an arbitrary "a" , just input atan(a) insted of a, and if you are not in RAD mode, remember to press →RAD. (ex. from 2 to infinite, instead of just pressing 2, press 2 and ATAN if you are already in RAD mode, otherwise press 2, ATAN and →RAD.

IMPORTANT NOTE: the program switches to RAD mode. Remember to restore DEG mode if you use your calculator this way.

Enjoy!