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

10232021, 11:46 AM
(This post was last modified: 11232021 08:43 AM by Roberto Volpi.)
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(35S) Integration with infinite limits ∫+∞, ∞
In the HP35S manual at E3 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 HP32S"" 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! Put a calculator into your life! 

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