tabular integration

(fg)' = fg'+gf' ( ' = d/dx)

 

therefore  

int fg' dx = fg - int f'g dx + const.

 

=>

 

int fg dx = fg_(1) - int f^(1)g_(1) dx (*)

where g_(1) = int g dx, f^(1) = d/dx f.

 

using (*) repeatedly,

 

int fg dx = fg_(1) - int f^(1)g_(1) dx

               = fg_(1) - { f^(1)g_(2) - int f^(2)g_(2) dx }

               = fg_(1) - f^(1)g_(2) + int f^(2)g_(2) dx

               = fg_(1) - f^(1)g_(2) + { f^(2)g_(3) - int f^(3)g_(3) dx }

               = fg_(1) - f^(1)g_(2) + f^(2)g_(3) - int f^(3)g_(3) dx 

and so on.

this is so-called tabular integration.

 

from the above, thinking of the table as below.

    f           g

    f^(1)    g_(1)

    f^(2)    g_(2)

    f^(3)    g_(3)

it is a very important method of integrals.