hello and welcome to my secret math tutor in this video we're going to take
on tabular integration and if you've never seen this technique before I think
you're in for quite a treat let's talk a little bit more about it. Tabular
integration is really just an organized way of doing integration by parts so if
you're not familiar with that method or you're not really quite sure what I'm
talking about check out my other video on what integration by parts is and how
to do it now since it really is integration by parts it really shines
for those problems where you have to do integration by parts multiple times
there are a few problems where you might have to do it over and over again and
this this table really allows you to take all that information and organize
it and be able to spit out an answer very fast now it also happens to work
for integrals that repeat themselves when doing integration by parts these
tend to be some very frustrating integrals but this method still works
just fine for them and of course the really interesting part is since it
really is just integration by parts in the background you can use this for
anytime you need to do integration by parts and you can stop it at any time
which really just gives you a simpler integral than what you started with all
right so that's enough talking about this method let's go ahead and give it a
try all right in this first problem we're
just going to get a feel of how the method works and when doing integration
by parts the first key step is identifying a piece where you can take
the derivative maybe something like this X cubed and another piece where you can
easily take the antiderivative like e to the X now what we're going to do with
each of these pieces is we're going to put them into our table so the
derivative piece will go on to the left side here so e cube or X cubed and the
antiderivative piece the piece where we want to take the antiderivative will go
on the right side and just to keep these two organized and and so I remember
which is which I'm going to write a little D over DX on the left column and
I'm gonna put a little integral symbol on the right column that way I know I
should be taking derivatives on this side anti derivatives over here alright
let's go ahead and work on that left side like I said we're just going to
start taking derivatives so the derivative of x cubed is 3x squared the
derivative of 3x squared is 6x the derivative of 6x is 6 and the
derivative of 6 goes to 0 so I've simply just taken a bunch of derivatives on the
other side we're going to take anti derivatives so we want to think
backwards antiderivative of e to the X is e to the X so that's nice and it
looks like it's just the same since the antiderivative of that is also e to the
X and all the way down now I only want to do as many rows as I have rows over
here so since this went off to 0 I'll go ahead and stop it
at that last row right there I won't necessarily continue going on all right
so this table has all the information we really need for writing down our answer
we just now have to collect it out of there how should we do that well we're
gonna connect the left column with the right column but we're going to do it
along a diagonal so X cubed I'll connect it with EDX over here 3x squared all
connected to this one over here I'll keep connecting them along that
diagonal until I've run out of terms now there's nothing left you'll see why I'm
not too worried about that 0 when we start getting into a little bit later
now that I have how these are connected we want to see what sign they're
connected with the first two are connected with a plus sign and the next
two are connected with a minus sign and you want to keep alternating as you go
down the table so plus minus plus minus let's go ahead and put those in plus
minus plus minus all right now it's time to write down our answer and here's how
we do that we take our pieces that are connected so in this case the X cubed
and the e to the X and we write them down we put the sign in front that
corresponds to the sign of how they're connected so in this case these I could
imagine like a little plus sign out front but I'm not going to go ahead and
write that on to the next piece 3x squared and e to the X yep that's how
those are connected and they're connected with a minus sign next pieces
I have a 6x and e to the X these are connected with a plus sign on the last
two six and e to the X these are connected with a minus sign
all right and there's our answer as long as we remember to include the nice
little plus C on the end and we're done now notice how we essentially did
integration by parts this was the you and the DV piece we just did it over and
over and this table helped us keep track of that so we could immediately jump to
the answer let's look at a different problem and see how this works
one more time all right like before we want to identify the piece where we need
to take derivatives and that's going to be this piece here since if I take
derivatives long enough it will also eventually go to 0 the other piece we
want to think of taking its anti derivative this will not zero out but
you know those trigonometric functions they tend to cycle back so I think we'll
be ok all right let's write down these pieces they become the first row in our
table and away we go taking derivatives the derivative of two
x squared will be 4 X the derivative of 4 X goes to 4 and the derivative of 4 is
simply 0 so this one zeroed out again off on the right side here time to take
anti derivatives so we have to think carefully all right what is the
antiderivative of cosine well that goes to sine antiderivative of sine negative
cosine and the antiderivative of negative cosine negative sine now that
we have those connect them along the diagonal and now we can write down the
signs of how they're connected remember these start with plus but then they
alternate so you'll go plus minus plus all the way down until you've run out of
stuff all right that looks good now let's go ahead and write down what we
have so the first two pieces here looks like we have a 2x squared plus 1
multiplied by a sine of X plus sign in front looks good next two pieces 4x
multiplied by a negative cosine of X and these are connected with a minus sign so
- and the last two pieces here looks like we have a 4 and a negative sine of
X connected with a plus sign all right now technically that is our
answer but it's always nice to take you know an additional step and connect
these double negative signs in there let's see 2x squared plus 1 multiplied
by sine of X negative times a negative will give us a positive so 4 X cosine of
X minus 4 sine of X put on our little plus C and now this guy is done now this
technique does work good even if your integral tends to repeat itself let's go
ahead and see what that would look like using this technique alright so as
before we want to identify pieces where we can take the derivative and
antiderivative and in this particular integral it doesn't really matter which
one we're going to choose for derivative or antiderivative so let's go ahead and
just choose e to the 2x for our derivative piece and we'll choose sine
of X for the piece where we will take anti derivatives all right into our
table so I have either 2 X and I have sine of X right so on the left side it's
time to take some derivatives derivative here using our chain rule is 2 e to the
2x now I'm thinking of the derivative of this and again I'm using the chain rule
so I'll get 4 e to the 2x and the derivative of again bring down that to
let's see 8 e to the 2x and you can see this one is not zeroing out if I were to
continue this process it just keeps going over and over so let's see let's
do one more time 16 e to the 2x dot dot keeps going alright onto the right side
for the right side we're doing anti derivatives so the antiderivative of
sine is negative cosine the antiderivative of negative cosine now
we're at negative sine antiderivative of that cosine of X and
the antiderivative of cosine sign and of course this would also continue if we
really were feeling ambitious now what you want to do in this case is comb over
your table and look for two pieces next to each other
that involve your e to the 2x and your sine of X of course it will be the first
row on a table but we're looking at for the very next instance that has those
two pieces now it doesn't matter if they have constants in front of them we
really again just want the e to the 2x and the sine of X and we can see that
that happens right here in the third row here's my e to the 2x and there's my
sine of X yes they have a 4 and a negative sign but I'm not really too
worried about those will be just fine so I'm drawing this dotted line here and
what that means is I'm going to stop this process after that point I'm not
really worried about anything that comes after it
all right time to connect our pieces like before we will connect along the
diagonal and when we get to this last one I don't really want to cross this
dotted line so I'm going to connect things straight across so it's kind of a
special one watch what I do with this one when I start writing out the answer
all right so we have our connections we want to connect them using plus
the other one is minus and plus
excellent now it's time to write down what we have see where we can go from
there so the first two pieces I have an e to the 2x multiplied by a negative
cosine of X of course those are connected with a little plus sign out
front that's good next two pieces to e to the 2x connected with a negative sign
these are connected with a negative science let's go ahead and put that in
and now the last two pieces this is that special one we still have them so I have
a four e to the 2x and I have a negative sign notice how I'm not connecting these
along the diagonal I'm connecting them straight across these are special
because they're inside of an integral we've stopped the process early so these
ones are not quite done yet and out in front of that integral we
will put a plus sign so we do have something a little bit
simpler than one we start with and of course we're choosing this example
because it has the similar piece on the inside let's go ahead and simplify our
signs and see exactly what's going on so let's see let's move that negative sign
out front
negative times a negative will give us a positive
here I'll move out the negative sign and we'll also move out that four so I'll
have a negative for e to the 2x sine of X DX and now you can really see it that
integral on the right side there is exactly the same as what we started with
but that's okay since it is the same I can go ahead and combine it together
with my original integral on the left side and we'll be alright so here's my
original here.i have here.i have subtracting four
of them will add forward to the right side and we'll add forward to the left
side let's see there's only one of them over here so four plus one will give us
five of our original integral
all right that will take that one put it over here everything else is exactly the
same
and now we are almost home free the last step of this really is just divide by
five and now we can see the value of the integral we were we were originally
interested in so there's our original integral everything divided by five so I
have a negative one-fifth e to the 2x cosine of X plus a 2/5
e to 2x sine of X and of course don't forget our plus C now it's just one last
example to show you that you can really use this for any problem that uses
integration by parts let's take care of this one
this one involves an X to the three halves x a natural log of X so if you
were to try and do derivatives and anti derivatives none of these really cancel
out or go to zero so we're gonna have to stop that process early also none of
these necessarily cycle back on themselves like trigonometric functions
will so watch why I stopped it early and why it really does help us alright so I
need a piece to take derivatives of and in this particular problem its nicest to
take derivatives of natural log that's because its derivative is just 1 over X
in terms of anti derivatives well take anti derivatives of X to the three
halves alright so we've identified our pieces let's go ahead and put them into
our table there they are so on the left side here let's start taking derivatives
derivative of natural log is 1 over X or since we'll be taking lots of
derivatives we could write this as X to the negative 1 alright another
derivative would bring down that negative 1 reduce the power so I have
negative x to the negative 2 another derivative these would multiply so I had
2 X to the negative 3 alright and of course this continues on but we'll go
ahead and at least do a few all right anti derivative over here we want to
think of adding 1 to the power and then dividing by that new power of course
when we divide by a fraction we want to flip and multiply so we'll get 2/5 X to
the 5 halves let's do another antiderivative
add 1 to the power divided by that fraction so the same as flipping and
multiplying that be 4 thirty-fifths and one more antiderivative just for good
measure so there's that 9 halves imagine flipping that over multiplying it by
four-fifths we would get 8 all divided by 315 all right even though this is
neither side is going towards 0 I can see that if I were to stop this what 2
pieces might be put together so after this the second row here if I were to
put the X to the negative 1 with the X to the 5 halves I'd only be left with
X's inside of my integral and that's actually a pretty good place to stop so
let's go ahead and put in our dotted line right here to show that I'll only
be really needing this table for the first two rows we'll connect them along
the diagonal and since I have my dotted line we'll just connect these straight
across so in a way this is like doing integration by parts only once
there's my plus sign and there's my minus sign all right let's write out
what we have so the first part I'll have the natural log of X multiplied by a 2/5
X to the 5 halves next two pieces here these will end up inside of an integral
and we'll subtract the integral away that's because of this negative sign
here and of course we're connecting them across let's see X to the negative one x
2/5 X to the 5 halves alright excellent so I start with this integral we have
something a little bit nicer now we just have to clean everything up let's go
ahead and rearrange the order of these I don't want anyone else to think that
maybe this is inside of my natural log it is not so let's see we have 2/5 X to
the 5 halves natural log of X minus combining these on the inside let's take
out that 2/5 we'll get X to the three halves
all right and of course that's a perfectly nice integral we can take the
antiderivative of that with no problems so let's go ahead and do just that the
antiderivative of X to the three halves we would add one to the power so five
halves divided by that new power two-fifths I will go ahead and put a
plus C on it right there all right let's go ahead and clean it up in just one
more step and then I think this one is great so we have 2/5 X to the 5 halves
natural log of X minus 4 25th X to the 5 halves plus C and then it's
done so hopefully that helps you out and you learn a brand new technique for
taking care of integration by parts one that really organizes your information
in a much better way if you'd like to see more videos please visit my website
at mysecretmathtutor.com thanks again