BAM! Mr. Tarrou! In this lesson we are going to help you deal with some of those problems
in your homework involving integration by parts that really take a lot of work. Now
maybe your textbook warned you about these type of problems and if they did then you
probably wouldn’t be watching this lesson unless you just want a few more examples,
but I have seen some calculus books that don’t sort of give you the shortcut method to deal
with some of these integration problems, integration by parts problems you identify u, you identify
dv. Boom... you are done in like a couple of lines and there are some that take a few
lines or maybe have to take a couple of steps of integration by parts to complete the problem...but
then it’s just... there’s some of them you do get to an end. But then you have to
do integration by parts three, four, five times and that can be like a one problem could
be just whole page with the work. So if you are realizing that you are going to need to
do multiple steps of integration by parts then a technique called the Tic Tac Toe method,
which was what they called it in the movie 'Stand and Deliver', awesome movie if you
have not seen it or even heard about it.. I am getting a little old! Or something more
officially called the Tabular Method. And when you are going to identify, hopefully,
when this technique, there's a couple of names up here, but it’s really just one technique,
is going to help you streamline your work and get the answer much quicker. You are going
to be looking for a factor of you integrand which is a polynomial. Okay our first example,
we are going to do in this lesson, if I am talking too much, you can go to the description
and skip to the next one... or I am starting this one now. The indefinite integral of x^2
times the sin (3x) dx. Now we are also with the...all my integration by parts lesson,
we are letting the LIATE acronym to help us to identify what factor to set equal to u,
that factor of our integrand which of course needs to be simpler when we take its derivative.
And do we have any logarithmic functions in our integrand? No. Do we have any inverse
trig functions? No. Do we have any algebraic expressions? Why yes we do! And really that’s
one of the conditions of us thinking that maybe this tabular method or tic tac toe method
is going to streamline our integration by parts problem. So then you have some trig
functions and exponential functions. So our algebraic portion of this example is of course
going to be x^2, the rest of the factors or the rest of our integrand is going to be our
dv. So dv is equal to the sin (3x) dx and what you want to do, you are also going to
have some sign changes as you go down this tabular method or tic tac toe method. And
what we are going to do is.. this is sort of like our headers if you will of this table
that we are going to make and in this first column we are going to be taking the derivatives
of u until we get down to zero. So the derivative of x^2 is going to be 2x. The derivative of
2x, I don’t want pointing up here now, is going to be 2 and then the derivative of a
constant of course is going to be zero and that right there is where you stop. We are
going to stop where that column of the derivatives of u becomes zero and that’s going to be
the last line of our table that we are going to use to figure out our answer. Now in this
second column, we need to integrate dv. So if v, I am not going to have any room for
scratch work so I am going to put it here but then erase it, we have v is going to be
equal to the indefinite integral of the sin(3x)dx. Of course this is pretty simple. Of course,
by now, I hope to integrate sine, that’s going to be negative cosine. The integral
of cosine is negative cosine. But it is not the integral of sine, it is the integral of
sin (3x). We have an inside function here. And when I first introduce all this.. these
basic integration rules, I do a u substitution and I get all the references of x out, replace
them all with u and I still use that a lot when I do integration, but if it’s a fairly
simple problem, we can just complete the..the chain rule pattern so we can then just integrate
without going through a whole u substitution process and removing all the references to
x. so our inside function is 3x. The derivative of 3x is equal to 3. Now it means that I need
to introduce a multiple of 3 and if I am going to introduce a multiple of 3 then I need to
balance that out with a division of 3. So I am effectively just multiplying by 1. And
now I have the chain rule pattern where it’s like..this is like... this is my outside function,
the derivative of my outside function with the original inside function times the derivative
of the inside function and that means that if we have v is equal to 1/3 the indefinite
integral of 3 sin(3x) dx then it’s also, after the integration process, -1/3 cos(3x).
And so that’s going to be the first entry in the second column. now at the end of this
column I am going to indicate that we are going to be sort of multiplying by a positive
or positive one if you will, so this is going to be our sign over here, I could write, I
could just write a circle with a sign or an addition sign in it. or maybe like a, I don’t
want to write +1, cause I don’t want to make it seem like we are going to be adding
1 or actually this is going to be a sign change or effectively be multiplying by 1 or -1.
That’s going to alternate as we work our way down these columns. But just to speed
up the video a little bit, let me step off and reveal these next couple of lines right
now. So we see in the second column that its dv and its antiderivatives. So we took the
antiderivative once and repeated that process a couple of more times. We have our sign changes.
Now the first line stats off with sort of a multiple of +1 if you will and it alternates
as you go down the list and you just keep going down again until you get that u giving
you a derivative of zero. Now it’s kind of acute to say that we are going to do this,
you know what could be, a very lengthy problem by just the Tic Tac Toe method, is catchy!
I am not exactly sure what all these letters that are actually in Tic tac toe are supposed
to represent. The only thing that I caution, you will see this in the next example, is..
I don’t think that because its tic tac toe, that you are necessarily going to have three
steps or rows in this table to be able to do the problem. If that were a higher degree,
it’s going to take more lines. But we are connecting the dots, tic tac toe. What are
we doing? We are multiplying along these branches. So we are going to have x^2 times -1/3 cos
(3x) and it’s positive here in the middle, at the end, excuse me. So it’s not going
to change the negative that’s in our middle multiplication. So we get -x^2 over 3 cos
(3x) and then we just keep doing that with each of our tic tac toes. 2x times -1/9 is
-2x over 9 and of course there’s sin(3x) there as well but then we have another multiple
of a negative out there so negative times negative is positive so we are going to have
plus 2x/9 times the sin(3x). And then 2 times 1/27 is 2/27 cos(3x), and we are multiplying
by +1 at the end and everything has been positive so far so we have +2/27 cos(3x). Now at this
point, we need to remember, as I am checking my answer to make sure that I wrote my notes
in there, that we are doing indefinite integration, so while there's no plus c's here, we effectively
have what could be, represented in our final answer so let’s make sure we put in our
constant of integration and I have kind of ran out of room so let me make sure I divide
that and that’s awesome answer. and one that I highly believe that your teacher is
going to be happy with but if you are checking your answers with a CAS calculator maybe or
you are checking your answers in the solution guide, that might not be the final answer
or version of how your answer looks. As I look at these two terms here, they both have
a factor of cos(3x). And focusing on these first and third term, we can divide out by
some kind of value that’s going to cancel out both of those denominators. And certainly
three times nine is equal to 27 so I can factor out a 1/27 and when I do that, we are going
to have, let’s see what we got here in the first term, we have the cos (3x) is out, we
have 1/3 divided by 1/27, remember... I don’t know if.. now that you are calculus class,
I am not sure you really need this now when you are factoring out a number, you dividing
out that number, right? so 1/3 divided by, no, there’s a negative there, so I have
a negative, but 1/3 divided by 1/27 is 1/3 times 27/1 cause that division by a fraction
is multiplying by the reciprocal , and 27/3 is going to be equal to 9, so we are going
to have -9x^2, now this is going to be a bit more obvious, if I have a denominator 27 or
1/27 and I factor that out, that's just going to leave us with +2 and of course our cos(3x)
is out front and we can’t do anything with this last- middle term so we have 2x/9 times
the sin(3x) plus c and that is the end of our first example. Now we are going to have
another example, it’s really the same process, I am just going to put it up here, I am not
even going to read it. I am just going to put another problem up here, give you about
five seconds to pause the video and try it on your own just to make sure you have the
process nailed down, then reveal the solution one step at a time, and make sure that you
can do this on your own when you come up to these kind of problems in your homework again.
BAM Alrighty then! So we took the derivative of that right down till we got to zero. And
I showed you on the first step of integrating this dv is equal to e^2x, just integrating
that over and over and over until we get to that last line where our derivative of u became
zero. and then started off our first line with a positive sign, always do that that
and then we are going to alternating those signs as we come down, again to our last row
and TIC TAC TOE TIC! You got to just keep going, it’s not three steps, it’s going
until that derivative of u comes out to be zero and then multiplying along all these
branches, we get 3x^3-2x times 1/2 e^2x minus 9x^2-2 times 1/4 e^2x plus 9/4 times x times
e^2x minus 9/8 e^2x plus c. Do not forget your integration, constant of integration.
Now , looking along both these terms, they both or they all, excuse me, have a factor
of e^2x and what I also did as I looked at these denominators and decided that, well
that is well there. And decided that that if I pull a 1/8 that would divide out that,
division of 8, or basically sort of yeah.. that would cancel out all of these denominators.
So indeed, 1/2 divided by 1/8 is 1/2 times 8 which is 4, 1/4 divided by 1/8 is 1/4 times
8 which is 2 and same thing here, we have 9/4 but that division of 4 getting divided
by 1/8 is multiplying by an 8, 8 divided by 4 is 2, they cancel out right? And 2 times
9 is equal to18. And of course we took that 8 out front. so doing that we distribute these
coefficients of 4 and -2 combine like, all our like terms and we get to a final simplified
answer of 1/8 e^2x times 12x^3-18x^2+10x-5 plus c. so hopefully that is what you got
if you paused the video and tried this on your own, if not I hope you can replay it
and kind of see where you messed up a little bit. I hope this helped. I am Mr. Tarrou!
BAM! Go do your homework!