sorry for hijacking this thread.

heys you know the spesh derivatives that are complicated, it can done be through 'anitderiving through derivatives right'?

is it true that it is possible to antiderive through complicated derivatives? is it way of vce level?

if its possible can any of you guys show me an example of it?

yup exactly what brightsky said, this is just integration by parts, you don't need to know this for spesh either, however it's not too hard to learn, it's just derived from the product rule, wikipedia is nice like brightsky said, however just for heck of it, i'll provide an example

basically to derive it, check wiki, you simply just antiderive the product rule, then after rearranging this is the result:

basically u and v are functions, ill show u an example and this will make more asense, note that here im stressing more about the application of this "formula" rather than going through rigorous details etc

eg

first we need to guess which function is u and which is dv, ie what i'm doing is basically "equating equations" (again im not stressing formality, simply showing you the mechanics) so what im doing is this:

we have the integration by parts rule

then we got our "equation" that we need to integrate, ie

so its like saying

so we have to guess, what is u and what is dv, ie, lets guess u = x and dv = sin(x)dx

yes if you're wondering, there's another guess we cudda taken, ie, u = sin(x) and dv = x dx [as you will see only one "pair" would work]

so if we let u = x and dv = sin(x) dx

then du = dx and v = -cos(x)

then all we gotta do is sub this into the formula!

so it becomes

but we know how to do

and so the rest is trivial

you go to try the other "guess" and you will see that the 2nd integral is not something we can integrate easily, so picking the first guess is better.

hopefully that makes a tiny bit more sense, note integration by parts just takes practise, there's no set method, you gain experience as you do more, so lets say

[note this is actually

]

here u can either do let

and dv = 1dx or u = 1 and

but it is very clear why the 2nd guess doesnt work!