WEBVTT
00:00:01.636 --> 00:00:04.446 A:middle
>> Welcome to the Cypress
College math review
00:00:04.446 --> 00:00:06.476 A:middle
on integration by parts.
00:00:10.706 --> 00:00:14.276 A:middle
Integration by parts is
a method that allows us
00:00:14.326 --> 00:00:16.946 A:middle
to integrate more
types of integrals.
00:00:17.616 --> 00:00:21.316 A:middle
Especially those consisting
of a product of two functions.
00:00:22.296 --> 00:00:25.736 A:middle
Let u and v be differentiable
functions
00:00:25.736 --> 00:00:33.636 A:middle
of x. I can write the
derivative of u either way.
00:00:35.736 --> 00:00:42.296 A:middle
Also, the derivative
of v. The differential
00:00:42.296 --> 00:00:44.136 A:middle
of u can be written this way.
00:00:45.606 --> 00:00:47.346 A:middle
And here's the differential
of v.
00:00:51.446 --> 00:00:52.596 A:middle
The product rule
00:00:53.516 --> 00:01:01.836 A:middle
for a derivative gives us u
prime times v plus v prime times
00:01:01.836 --> 00:01:08.886 A:middle
u. Integrating both sides of
this equation with respect to x.
00:01:26.196 --> 00:01:36.236 A:middle
Gives us the antiderivative of
v. And then we have u prime dx.
00:01:36.236 --> 00:01:39.756 A:middle
Which, of course, from
what we have above, is du.
00:01:41.796 --> 00:01:48.796 A:middle
And then we're integrating u.
And then v prime dx gives us dv.
00:01:48.846 --> 00:01:51.946 A:middle
If we solve for this integral.
00:01:57.006 --> 00:02:02.776 A:middle
We have the formula for
integration by parts.
00:02:07.346 --> 00:02:10.786 A:middle
Once you've determined that
you're going to use integration
00:02:10.786 --> 00:02:13.486 A:middle
by parts to evaluate
your integral.
00:02:14.886 --> 00:02:19.056 A:middle
You need to determine
which factor will be u.
00:02:19.656 --> 00:02:21.456 A:middle
Which you will be
differentiating.
00:02:22.246 --> 00:02:26.336 A:middle
And which factor will be dv,
which you will be integrating.
00:02:27.366 --> 00:02:31.216 A:middle
For dv, choose the
most complicated part
00:02:31.216 --> 00:02:35.616 A:middle
of the integrand that fits
a basic integration rule.
00:02:37.306 --> 00:02:40.066 A:middle
When choosing the
factor that will be u,
00:02:40.936 --> 00:02:49.036 A:middle
use the acronym L.I.A.T.E.
Which stands for logarithmic,
00:02:49.906 --> 00:02:56.836 A:middle
inverse trig, algebraic, trig,
or exponential -- in that order.
00:02:57.736 --> 00:03:01.226 A:middle
In other words, if you have
a factor that's logarithmic.
00:03:01.606 --> 00:03:05.056 A:middle
For sure, that factor is u.
00:03:06.046 --> 00:03:10.666 A:middle
If you have a factor that's
inverse trig, make that u.
00:03:13.786 --> 00:03:16.546 A:middle
Recall once again
what our formula
00:03:16.546 --> 00:03:18.486 A:middle
for integration by parts is.
00:03:20.626 --> 00:03:24.776 A:middle
Now, a lot of instructors
will allow you
00:03:24.776 --> 00:03:27.316 A:middle
to use a tabular method.
00:03:28.066 --> 00:03:34.786 A:middle
With a tabular method,
we set it up like this.
00:03:34.876 --> 00:03:36.226 A:middle
I've seen other ways to do it.
00:03:36.286 --> 00:03:37.596 A:middle
But something like this.
00:03:38.076 --> 00:03:39.856 A:middle
Which part do you differentiate?
00:03:39.996 --> 00:03:44.566 A:middle
You differentiate the factor
u. Which part do you integrate?
00:03:45.106 --> 00:03:47.516 A:middle
You integrate the
factor dv, yes?
00:03:48.656 --> 00:03:52.656 A:middle
When you differentiate
u, you get du.
00:03:53.796 --> 00:03:58.806 A:middle
When you integrate dv, you
get v. And then, of course,
00:03:59.136 --> 00:04:01.176 A:middle
because of the formula
that we have here.
00:04:01.176 --> 00:04:04.116 A:middle
We have to tack on
the plus and the minus
00:04:04.116 --> 00:04:05.736 A:middle
to get the appropriate answers.
00:04:06.076 --> 00:04:08.976 A:middle
So this original integral.
00:04:14.566 --> 00:04:22.896 A:middle
Is equal to this product,
minus this integral.
00:04:23.336 --> 00:04:26.446 A:middle
Now, did that really
save you time?
00:04:26.916 --> 00:04:30.266 A:middle
No. What's nice about
the tabular method,
00:04:30.676 --> 00:04:35.576 A:middle
is you can continue to do
this process numerous times.
00:04:35.726 --> 00:04:38.556 A:middle
Numerous steps over
and over again.
00:04:39.006 --> 00:04:45.486 A:middle
And often, integration by parts
has to be done numerous steps.
00:04:48.266 --> 00:04:50.996 A:middle
U substitution does not
work on this problem.
00:04:52.836 --> 00:04:59.326 A:middle
And since we have a logarithmic
factor, u would be natural log
00:04:59.326 --> 00:05:05.276 A:middle
of x. So dv is x squared dx.
00:05:07.566 --> 00:05:12.436 A:middle
We differentiate
to get 1 over x dx.
00:05:13.766 --> 00:05:18.006 A:middle
We integrate to get
x cubed over 3.
00:05:19.536 --> 00:05:27.976 A:middle
So our original integral
is equal to u times v.
00:05:33.076 --> 00:05:41.646 A:middle
Minus the integral of v du.
00:05:50.976 --> 00:05:52.176 A:middle
Pull out the 1/3.
00:05:53.266 --> 00:05:57.976 A:middle
We cancel, and now we just
have x squared to integrate.
00:06:03.696 --> 00:06:07.256 A:middle
So to integrate x squared
using the power rule.
00:06:07.256 --> 00:06:11.106 A:middle
We would get x cubed
over 3 times the other 3
00:06:11.106 --> 00:06:12.736 A:middle
in the denominator, gives me 9.
00:06:14.396 --> 00:06:15.326 A:middle
We have our answer.
00:06:17.636 --> 00:06:19.976 A:middle
I'm going to show
the same problem done
00:06:19.976 --> 00:06:22.876 A:middle
over using the tabular method.
00:06:25.506 --> 00:06:27.706 A:middle
So in this column, we're
going to differentiate.
00:06:28.206 --> 00:06:30.126 A:middle
And in this column,
we're going to integrate.
00:06:31.166 --> 00:06:33.976 A:middle
So looking at natural
log and x squared.
00:06:33.976 --> 00:06:37.256 A:middle
We're definitely going to
differentiate natural log.
00:06:37.256 --> 00:06:39.766 A:middle
Since we don't know how
to integrate natural log.
00:06:40.856 --> 00:06:43.556 A:middle
So x squared will be the
factor that we integrate.
00:06:45.206 --> 00:06:49.546 A:middle
The derivative of natural log
is 1 over x. The antiderivative
00:06:49.546 --> 00:06:52.936 A:middle
of x squared is x cubed over 3.
00:06:54.736 --> 00:06:57.746 A:middle
Then, because of the formula,
you put on your signs.
00:06:57.746 --> 00:06:58.956 A:middle
So plus, minus.
00:06:58.956 --> 00:07:01.966 A:middle
And then, if this
process continued on
00:07:01.966 --> 00:07:05.786 A:middle
and you did integration by
parts through multiple steps.
00:07:05.786 --> 00:07:08.876 A:middle
You would continue this
process down with the signs.
00:07:09.976 --> 00:07:12.836 A:middle
And you'd keep differentiating,
keep integrating.
00:07:13.706 --> 00:07:14.966 A:middle
So the original integral.
00:07:19.506 --> 00:07:22.416 A:middle
Is equal to -- this
is the u times v part.
00:07:22.416 --> 00:07:26.756 A:middle
So on the diagonal is u times
v. So that is a product --
00:07:31.856 --> 00:07:36.896 A:middle
-- Minus. And horizontal
is the integral.
00:07:45.456 --> 00:07:48.136 A:middle
And you can see, that's the
same problem I just did.
00:08:03.346 --> 00:08:06.466 A:middle
And here we go.
00:08:06.466 --> 00:08:08.476 A:middle
On this problem you
have a choice.
00:08:09.406 --> 00:08:12.326 A:middle
You can let u be the
exponential function,
00:08:12.746 --> 00:08:14.496 A:middle
or the trigonometric function.
00:08:15.096 --> 00:08:16.176 A:middle
It really doesn't matter.
00:08:16.896 --> 00:08:19.346 A:middle
I'll let u be the
trigonometric function.
00:08:20.976 --> 00:08:23.716 A:middle
So we'll let u be cosine of 3x.
00:08:26.006 --> 00:08:33.626 A:middle
Which means that du will be
negative 3 times sine of 3xdx.
00:08:33.726 --> 00:08:42.416 A:middle
Dv then, will be e to the 4x dx.
00:08:45.516 --> 00:08:52.896 A:middle
And v then, will
be 1/4 e to the 4x.
00:08:54.636 --> 00:08:56.276 A:middle
So our original integral.
00:09:04.146 --> 00:09:14.356 A:middle
Is equal to u times v. Which is
1/4 e to the 4x cosine of 3x.
00:09:16.016 --> 00:09:17.846 A:middle
Minus the integral of v du.
00:09:19.456 --> 00:09:21.106 A:middle
I have a double negative.
00:09:22.576 --> 00:09:36.376 A:middle
So I have plus and then 3/4,
e to the 4x sine of 3xdx.
00:09:37.856 --> 00:09:42.406 A:middle
But once again, I have another
integral that I'm going to have
00:09:42.496 --> 00:09:45.616 A:middle
to use integration
by parts to do.
00:09:46.476 --> 00:09:47.816 A:middle
So we're starting over.
00:09:48.406 --> 00:09:55.986 A:middle
This time, u is equal
to sine of 3x.
00:09:56.586 --> 00:10:04.436 A:middle
Du is equal to 3 cosine of 3xdx.
00:10:05.066 --> 00:10:10.976 A:middle
Dv, e to the 4x dx.
00:10:11.466 --> 00:10:17.086 A:middle
Which gives us v is
equal to 1/4 e to the 4x.
00:10:19.676 --> 00:10:21.726 A:middle
So our original integral.
00:10:27.926 --> 00:10:30.646 A:middle
Is equal to our original
u times v.
00:10:36.446 --> 00:10:40.576 A:middle
Plus 3/4 times our new integral.
00:10:41.576 --> 00:10:49.056 A:middle
Our new integral has the new
u times v, which is 1/4 e
00:10:49.056 --> 00:10:57.866 A:middle
to the 4x sine of 3x,
minus the integral of v du.
00:10:59.296 --> 00:11:06.736 A:middle
Which, in this case, would be
3/4 times the integral of e
00:11:06.736 --> 00:11:12.366 A:middle
to the 4x cosine of 3xdx.
00:11:13.326 --> 00:11:18.296 A:middle
Please note that this
integral and the integral
00:11:18.296 --> 00:11:19.606 A:middle
on the left-hand side.
00:11:19.606 --> 00:11:23.026 A:middle
Which was my original integral,
are multiples of each other.
00:11:23.356 --> 00:11:25.476 A:middle
Those integrals are the
-- it's the same integral.
00:11:25.476 --> 00:11:27.846 A:middle
But we have a different
constant in front of them.
00:11:29.096 --> 00:11:30.976 A:middle
Alright, let's take a
look at those constants.
00:11:36.116 --> 00:11:39.666 A:middle
So let's deal with the
coefficients in front of these.
00:11:44.266 --> 00:11:46.146 A:middle
So here I have 3/16.
00:11:53.716 --> 00:11:57.306 A:middle
And here I have negative 9/16.
00:12:06.336 --> 00:12:10.936 A:middle
So on the left-hand side, I
have 1 times this integral.
00:12:11.456 --> 00:12:12.736 A:middle
On the right-hand side,
00:12:12.736 --> 00:12:16.656 A:middle
I have negative 9/16
times this integral.
00:12:17.766 --> 00:12:24.706 A:middle
So I'm going to add 9/16
times this same integral
00:12:26.576 --> 00:12:30.256 A:middle
to both sides.
00:12:41.536 --> 00:12:54.296 A:middle
Yes? When I do that, I get 25/16
times that integral is equal
00:12:54.296 --> 00:12:56.266 A:middle
to these two terms here.
00:13:14.046 --> 00:13:16.596 A:middle
Then I simply multiply
both sides
00:13:16.596 --> 00:13:18.846 A:middle
by the reciprocal of 25/16.
00:13:28.046 --> 00:13:35.276 A:middle
So once again, I'm
multiplying by 16 over 25.
00:13:36.016 --> 00:13:39.536 A:middle
And I would get 4 over 25.
00:13:40.046 --> 00:13:44.706 A:middle
Once again, over here, I'm
multiplying by 16 over 25.
00:13:44.706 --> 00:13:46.616 A:middle
Both sides of the
equation by that.
00:13:48.386 --> 00:13:51.536 A:middle
Yes. So distribute it to there
and you distribute it to here.
00:13:52.736 --> 00:13:54.726 A:middle
So I have 4 over 25 here.
00:14:01.046 --> 00:14:02.906 A:middle
And here I get 3 over 25.
00:14:08.186 --> 00:14:11.836 A:middle
And then, of course, plus
c. And we have our answer.
00:14:15.666 --> 00:14:18.876 A:middle
We're going to do this problem
again using the tabular method.
00:14:20.306 --> 00:14:25.756 A:middle
We also want to talk about the
number of steps of integration
00:14:25.756 --> 00:14:28.426 A:middle
by parts it takes
to do a problem.
00:14:30.426 --> 00:14:32.326 A:middle
So we differentiate
what's on the left
00:14:32.326 --> 00:14:34.126 A:middle
and we integrate
what's on the right.
00:14:36.226 --> 00:14:38.916 A:middle
So we have our steps
that we're --
00:14:39.986 --> 00:14:42.896 A:middle
our functions that we're
differentiating on the left.
00:14:43.336 --> 00:14:44.596 A:middle
Integrating on the right.
00:14:44.596 --> 00:14:47.356 A:middle
We put on our signs.
00:14:47.356 --> 00:14:49.346 A:middle
So we're going down through.
00:14:49.866 --> 00:14:52.136 A:middle
And how do we know when to stop?
00:14:52.776 --> 00:14:58.896 A:middle
The original integral is
equal to this product.
00:14:59.766 --> 00:15:04.746 A:middle
Plus this product, plus this
product, and so on and so forth.
00:15:05.256 --> 00:15:07.336 A:middle
Plus this integral.
00:15:09.126 --> 00:15:18.956 A:middle
So either you can
integrate this.
00:15:24.546 --> 00:15:26.526 A:middle
Or, it is a multiple --
00:15:33.776 --> 00:15:39.276 A:middle
-- Of the original integral.
00:15:42.496 --> 00:15:43.666 A:middle
Like the last problem.
00:15:46.096 --> 00:15:50.706 A:middle
That's how many steps it takes
to do integration by parts.
00:15:51.076 --> 00:15:52.436 A:middle
That's when you know
you're done.
00:15:55.786 --> 00:16:03.736 A:middle
So let's differentiate cosine
3x and integrate e to the 4x.
00:16:05.046 --> 00:16:10.036 A:middle
So the derivative is
negative 3 sine of 3x.
00:16:10.426 --> 00:16:14.636 A:middle
The antiderivative
is 1/4 e to the 4x.
00:16:15.206 --> 00:16:16.086 A:middle
Are we done?
00:16:17.636 --> 00:16:20.196 A:middle
Can we integrate
this bottom row?
00:16:21.026 --> 00:16:25.496 A:middle
No. Is this bottom
row a multiple
00:16:25.496 --> 00:16:26.756 A:middle
of the original integral?
00:16:27.456 --> 00:16:29.536 A:middle
No. Keep going.
00:16:31.146 --> 00:16:32.976 A:middle
Differentiate again.
00:16:41.306 --> 00:16:42.586 A:middle
Alright, let's see
if we're done.
00:16:43.426 --> 00:16:45.606 A:middle
Can we integrate
this bottom row?
00:16:46.346 --> 00:16:50.446 A:middle
No. Is this bottom
row a multiple
00:16:50.446 --> 00:16:52.086 A:middle
of my original integral?
00:16:53.096 --> 00:16:54.746 A:middle
Yes. We're done.
00:16:55.866 --> 00:17:01.066 A:middle
Tack on the signs,
let's do this.
00:17:02.266 --> 00:17:03.946 A:middle
So the original integral --
00:17:11.156 --> 00:17:14.866 A:middle
-- Is equal to our
-- this product.
00:17:24.156 --> 00:17:25.496 A:middle
Plus this product.
00:17:26.806 --> 00:17:36.446 A:middle
So 3/16 u to the 4x sine
of 3x, plus this integral.
00:17:36.446 --> 00:17:42.426 A:middle
So that's minus 9/16
times the integral of e
00:17:42.426 --> 00:17:44.666 A:middle
to the 4x cosine of 3x.
00:17:48.446 --> 00:17:55.846 A:middle
So I'm going to add 9/16 times
that integral to both sides.
00:18:09.416 --> 00:18:13.686 A:middle
So we get 25/16 times
the integral.
00:18:17.826 --> 00:18:18.856 A:middle
Just like we did before.
00:18:21.256 --> 00:18:23.106 A:middle
So we have the same answer.
00:18:25.606 --> 00:18:27.686 A:middle
Skip a step since we've
already done this problem.
00:18:51.126 --> 00:18:51.556 A:middle
There we go.
00:18:54.536 --> 00:18:57.736 A:middle
For multiple step
problems, there's nothing
00:18:57.736 --> 00:18:59.056 A:middle
like the tabular method.
00:19:02.896 --> 00:19:10.866 A:middle
So, since our algebraic function
will differentiate basically
00:19:11.176 --> 00:19:12.056 A:middle
into nothing.
00:19:13.186 --> 00:19:14.066 A:middle
We will do that.
00:19:15.946 --> 00:19:18.546 A:middle
Our trigonometric function
will just alternate
00:19:18.626 --> 00:19:20.376 A:middle
between sines and cosines.
00:19:21.916 --> 00:19:22.976 A:middle
We'll let it do that.
00:19:27.466 --> 00:19:30.506 A:middle
How far -- how do I, how
do I know how far to go?
00:19:31.816 --> 00:19:34.686 A:middle
Until it's disappeared
into nothing.
00:19:39.066 --> 00:19:40.796 A:middle
I didn't really have
to go that far.
00:19:41.356 --> 00:19:42.716 A:middle
How far did I have to go?
00:19:42.716 --> 00:19:44.796 A:middle
I had to go until
I could integrate.
00:19:45.336 --> 00:19:47.706 A:middle
So I could've done
one less step.
00:19:48.706 --> 00:19:53.576 A:middle
I could've integrated 6 cosine
x. But why not go ahead and go
00:19:53.576 --> 00:19:55.526 A:middle
down to just integrating zero.
00:19:56.666 --> 00:19:58.306 A:middle
Yeah? Nothing wrong with that.
00:19:59.386 --> 00:20:00.946 A:middle
So our answer is.
00:20:07.416 --> 00:20:17.176 A:middle
This product, negative x cubed
cosine x. Plus this product,
00:20:18.276 --> 00:20:24.126 A:middle
3x squared sine of
x. Plus this product,
00:20:25.096 --> 00:20:32.596 A:middle
6x cosine x. Plus this
product, negative 6 sine
00:20:32.596 --> 00:20:35.936 A:middle
of x. Plus this integral.
00:20:37.026 --> 00:20:38.396 A:middle
Now what are you integrating?
00:20:39.086 --> 00:20:40.786 A:middle
You're integrating zero.
00:20:42.206 --> 00:20:44.796 A:middle
What's the antiderivative
of zero?
00:20:47.406 --> 00:20:51.136 A:middle
C, yeah? It's the constant
of integration, yeah?
00:20:51.686 --> 00:20:54.646 A:middle
When you integrate
zero with respect to x,
00:20:54.646 --> 00:20:57.706 A:middle
you simply get constant
of integration.
00:20:58.356 --> 00:20:59.446 A:middle
Don't you?
00:20:59.566 --> 00:21:06.136 A:middle
It's just c. There's nothing
like the tabular method
00:21:06.446 --> 00:21:08.706 A:middle
for integration by
parts when you have
00:21:08.706 --> 00:21:10.886 A:middle
to do it multiple steps.
00:21:13.886 --> 00:21:18.396 A:middle
Here are some more problems
that you might want to do.
00:21:18.606 --> 00:21:19.766 A:middle
Pause the video.
00:21:20.476 --> 00:21:21.906 A:middle
Try them on your own.
00:21:23.136 --> 00:21:26.646 A:middle
Then restart the video
and check your answers.
00:21:31.816 --> 00:21:33.976 A:middle
Here are the answers to
the practice problems.