Let’s go back to our rather bulky looking problem:



In the “Tic Tac Toe” method we create a table, but we repeatedly differentiate one function while we correspondingly repeatedly integrate the other:

NO SIGN HERE - JUST RESTATE 
THE PROBLEM ON THIS LINE


The “Tic Tac Toe” expression comes from the “joining” of the similarly coloured rows, taking account of the sign operation corresponding to the derivative term:


Quite a bit faster don’t you agree?


The best way to make sure that you get the signs right would be to ‘group’ the terms without signs first and then add the signs according to their ‘natural sign’ and any changes made by the sign to be applied from the third column:



We have to establish the correct sign for each of the placeholders 1, 2 and 3 above.


1. From Tic Tac Toe:

is negative, but the sign on line 2 is also negative, therefore placeholder 1 will be “+”


2. From Tic Tac Toe:

is negative, but the sign on line 3 is positive, therefore placeholder 2 will be “-“


3. From Tic Tac Toe:

is positive, but the sign on line 4 is negative, therefore placeholder 3 will be “-“


FINAL RESULT:


This integral took quite some time ‘long hand’ leading to the result, passing through several ‘sub integrals’ on the way there, but with the “tic-tac-toe method” we were there in a fraction of the time. I’m going to tabulate another example just to show you how quickly it can be done.

NO SIGN HERE - JUST RESTATE 
THE PROBLEM ON THIS LINE



We have to establish the correct sign for each of the placeholders 1, 2, 3 and 4 above.


1. From Tic Tac Toe:

is negative, but the sign on line 1 is positive therefore this expression overall will be (remain) "-"


2. From Tic Tac Toe:

is negative, but the sign on line 2 is also negative, therefore placeholder 1 will be “+”


3. From Tic Tac Toe:

is positive, but the sign on line 3 is also positive, therefore placeholder 2 will be “+”


4. From Tic Tac Toe:

is positive, but the sign on line 3 is negative, therefore placeholder 3 will be “-”


5. From Tic Tac Toe:

is negative, but the sign on line 2 is positive, therefore placeholder 4 will be “-”


FINAL RESULT:


>> Questions <<