[size=150][b]Tabular Method[br][br][/b]Suppose [img width=69,height=24]http://www.hyper-ad.com/tutoring/images/int_pa7.gif[/img] and [img width=65,height=24]http://www.hyper-ad.com/tutoring/images/int_pa8.gif[/img] . Then if we set up a table, differentiating f(x) as many times as it takes to get to zero and integrating g(x) as many times, we get[table][tr][td][b]D[/b][/td][td][b]I[/b][/td][/tr][tr][td][b][img width=20,height=20]http://www.hyper-ad.com/tutoring/images/int_pa9.gif[/img] (a)[/b][/td][td][img width=18,height=20]http://www.hyper-ad.com/tutoring/images/int_pa10.gif[/img][/td][/tr][tr][td][b][img width=21,height=16]http://www.hyper-ad.com/tutoring/images/int_pa11.gif[/img] (b)[/b][/td][td][b]+ [/b][img width=18,height=20]http://www.hyper-ad.com/tutoring/images/int_pa12.gif[/img] [b](a)[/b][/td][/tr][tr][td][b][img width=13,height=16]http://www.hyper-ad.com/tutoring/images/int_pa13.gif[/img] (c)[/b][/td][td][b]- [/b][img width=18,height=20]http://www.hyper-ad.com/tutoring/images/int_pa14.gif[/img] [b](b)[/b][/td][/tr][tr][td][b][img width=12,height=16]http://www.hyper-ad.com/tutoring/images/int_pa15.gif[/img][/b][/td][td][b]+ [/b][img width=18,height=20]http://www.hyper-ad.com/tutoring/images/int_pa16.gif[/img] [b](c)[/b][/td][/tr][/table][br]Notice how we alternate the signs in the “I” column. Then, multiplying rows with like letters (which you can skip on your own paper to help make this clearer - draw arrows instead from (a) to (a), etc.) gives the following directly as the anti-derivative:[br][br][img width=221,height=32]http://www.hyper-ad.com/tutoring/images/int_pa17.gif[/img][br][br]This method is much faster than the f-g method or the older u-v, especially for iterated (more than once) integrals by parts [/size][br]