Trouble is, some functions in I.b.p. take about 3 - 4 pages. Tabular integration for particular functions (especially if you later on do p.d.e.s) can be done in a small table and one line.
It's ultimately the same method. The tabular method just does a better job at organizing it, making it more compact, and is much easier to understand. Some teachers insist on you using the original formula, others encourage you to use the tabular method and let you use either method to do it.
It is best to not use any grouping symbols until you are finished integrating. Just carry the signs throughout the whole process and you will not mess up the signs. It really is simpler.
I found your video so much better than my teachers explenation, but i dont get how you at the 8.10 mark integrate e^x * cos(x) to e^x * cos(x)? like do you integrate twice or how does that work?
When you have a product of an exponential and a trig function, your integration by parts method is a looper. You would end up in an infinite loop, but you can stop the infinite loop by spotting the original integral within it. You then solve it as an algebra problem. For the example you gave, I assign e^x to be integrated, and cos(x) to be differentiated. I get the following table: S. . D . . . . . I + . cos(x) . e^x - . .-sin(x). . e^x + . -cos(x) . e^x Construct along diagonals, and construct the final row as an integral: +cos(x)*e^x - sin(x)*e^x - integral cos(x)*e^x dx Let this equal I, for the original integral. Notice that we also have the original integral inside the equation. We therefore can solve it as an algebra problem: I = cos(x)*e^x - sin(x)*e^x - I Add I to both sides: 2*I = cos(x)*e^x - sin(x)*e^x Solve for I: I = 1/2*(cos(x) - sin(x))*e^x Solution: 1/2*(cos(x) - sin(x))*e^x + C
