all right I just want to talk about how
we go about deriving the integration by
parts formula from calculus so a lot of
people have this memorized integral of
UDV equals UV minus the integral of VDU
people sometimes forget it for sure and
I just you know if you had to you know
obviously this is something that you
want to memorize it shows us a lot but
if you forget it here's a way to derive
it or maybe you know again you're just
interested as to where it comes from so
suppose we start with you know basically
what the integration by parts formula
kind of does is it it undoes the product
rule so suppose we have a product f of x
times G of X and we want to take the
derivative of that well the product rule
just says well take the derivative of
one of them so I'll take F prime of X
and then it says leave the other one
alone and then stick a plus sign and in
between and now we just kind of switch
them out so I took the derivative of F
we'll leave it alone I didn't take the
derivative of G originally but now I
will so that's just the goodö product
rule from from taking derivatives
usually in you know a first semester of
calculus well the idea is now if these
are all equal we can just calculate the
antiderivative of both sides so I can
take the antiderivative of the left side
and I can take the antiderivative of the
right side so f prime of x times G of X
I'm being a little sloppy I should stick
a DX in there when I do this in the
integral of f of X times G prime of X so
let me stick the DX in there as well so
DX okay the idea is though if you start
with a function so you start with
something f of X times G of X if you
take the derivative and then you
calculate the antiderivative you
basically get back to the original
function so f of X times G of X and ok
so I agree you do have to stick on a
plus C somewhere but we can stick on the
plus C on either side of this
equation so I'm not gonna put our plus C
in there just yet
well we I'm not really gonna do much of
anything to the right side except just
rewrite it here and since we haven't
integrated everything fully it's okay
that I don't have my plus C and they're
just yet okay
and actually the plus C will never show
up because we never actually completely
integrated so just in case that's one
thing that's bothering you so what I'm
gonna do now is I'm just gonna subtract
over this integral so I've got f of X
times G of X on the left side I'm just
going to subtract this whole integral
from both sides so if I subtract it over
well on the left side it'll go over as
negative f prime of x times G of X DX
well the only thing that you would be
left with now on the right side would be
this other integral f of x times G prime
of X DX and this is now the integration
by parts formula it's not the way you
usually see it written but it's
definitely equivalent and typically
you'll see books what they'll say is
they'll say something like well let's
let u be the function let's rename let's
rename f of X let's just call it U well
if we do like a little differential D U
it says we would get f prime of X DX
and likewise they'll say oh well let's
let V let's let that be the function G
of X which means DV would be G prime of
x times DX and now I'm just gonna plug
all this sort of relabeling into my my
formula so it says well f of X that's
what we're calling u G of X that's what
we're calling the minus the integral of
okay so we've got F prime x times DX
well F prime of X DX is going to give us
our D u G of X that's what we're calling
V and then on the right side our
function we've got the integral
of F of xdx but again F of X DX that's
what we're just calling excuse me our f
of X that's what we're just calling you
be careful here our G prime X DX that's
what we're calling DV and now we've got
our integration by parts formula it's
kind of got it written you know
flipped cuz normally we say the integral
of U DV equals UV minus the integral of
V D U but hey now we've got it that's
exactly where it comes from so again all
you're doing to get the integration by
parts formula it's just your undoing the
product rule as all you're doing so as
long as you remember the product rule
just stick some integrals in there if
you take the derivative and integrate
that you're basically back to where you
started again we don't have to put a
plus C because we've never fully
integrated both sides and then it's just
moving things over and just relabeling
and that's all there is to it so nothing
too mysterious just again you're just
doing the product rule undoing the
product rule and that gives us our nice
little integration by parts formula