WEBVTT
Kind: captions
Language: en
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break that apart
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then you figured this out right?
00:00:06.080 --> 00:00:06.760
and this
00:00:07.440 --> 00:00:11.000
is uhh 5x from 0 to 2.
00:00:11.820 --> 00:00:14.000
which is 10-0 which is 10.
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hopefully it came out 0.
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if i remember correctly
00:00:16.880 --> 00:00:18.760
but its memory so i could be wrong.
00:00:19.580 --> 00:00:21.960
okay because i think g(x) was 5
00:00:25.580 --> 00:00:30.700
cause you got that by doing the integral from 0 to 1 plus the integral from 1 to 2
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plus the integral from 2 to 3
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thats how you found the integral from 0 to 2 okay?
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this one
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okay
00:00:57.860 --> 00:01:00.320
so how do you do the derivative of an integral?
00:01:01.860 --> 00:01:04.500
first you'll plug the top function in so you'll get
00:01:05.000 --> 00:01:05.560
tangent
00:01:06.360 --> 00:01:10.700
of the ln(x^5)^2
00:01:11.940 --> 00:01:13.580
times 5x^4
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dont forget that part
00:01:16.400 --> 00:01:16.960
minus..
00:01:17.660 --> 00:01:24.740
tan(ln2x)^2x2
00:01:25.360 --> 00:01:25.860
ok?
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really could have left it like that you dont really have to simplify.
00:01:29.640 --> 00:01:31.400
but if you want to simplify
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you could do that
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dont do more than that dont show off your algebra.
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many of you in the act of simplifying take a right answer and turn it into a wrong answer.
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this makes it simpler for us to grade.
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dont simplify if you dont have to okay?
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professor T and i have the same philosophy which is this is good enough.
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that you know what youre doing.
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howd you like that second one the trigonometry one did you guys figure that out?
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wasnt that kind of fun?
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wasnt that hard.
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the answer to that was sinx i believe.
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plus c of course.
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alright for those of you who thought this was trigonometry
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i wrote this question so
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1-cos^2x is sin^2x right?
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this is sin^2x.
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so its going to cancel the sin^2x in the numerator.
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and just leave you with dx/secx.
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but wait
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sec in the denominator is cos in the numerator.
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so this equals sinx+c.
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there you go alright?
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this is the good stuff.
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we're going to learn how to integrate so for a lot of the next
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oh handful of classes
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we do whats called technique of integration.
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so this is the
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when you become physicists
00:04:04.700 --> 00:04:06.660
youre going to be really happy that you learned this.
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suppose you do not become physicists
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then you say why am i learning techniques of integration?
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well whats going to happen is
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physics
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computational biology, engineering
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economics
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statistics
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finance
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couple other areas
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you set up a problem, often you have to solve an integral
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because you figured something out, integrals are used to sum things.
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like we just did all those riemann sums right?
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so integrals
00:04:35.340 --> 00:04:38.040
among other things is just a way of adding up a lot of stuff.
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you want to add up a lot of stuff and the equation, you integrate it that gives you the sum.
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okay? riemann sum is back umm
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so you'll set it up, and then you look at the integral and you say
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i have no idea what the solution is.
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if it was derivative
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you could take the derivative but unfortunately
00:04:54.500 --> 00:04:57.380
you can differentiate just about anything
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but you cannot integrate just about anything.
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lots of things cant be integrated.
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its kind of like you smashed it but you cant always put them back together.
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so um
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youll be able to - but now we teach you how to integrate a variety of things.
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so, in fact the plus c, just because you know
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if you have x^3 you could find the derivative as 3x^2
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but if you know the derivative is 3x^2
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you dont necessarily know that the original function is x^3.
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theres lots of other functions it could be.
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some things cannot be integrated. they are
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not integrable.
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or theyre not, or theyre not very easily integrable.
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and you learn as you move up into higher math
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how to get really good approximations.
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like when you plug your answers into Wolfram Alpha
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as if we dont know about that
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okay wolfram alpha can give you an approximation
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as many digits long as you want.
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it'll spit out 200 digits if you ask it to.
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okay? and usually you dont need more than about 4.
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so
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first major technique is something called integration by parts.
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remember
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product rule
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product rule says that if you have 2 functions multiplied together
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the u and the v
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okay
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the derivative of that
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u times the derivative of v
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plus v times the derivative of u
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to use a little shorthand.
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udv+vdu
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so if we were to sorta integrate that
00:06:30.880 --> 00:06:32.500
you would get u times v
00:06:33.380 --> 00:06:37.720
the integral of udv + the integral of vdu.
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so you do a little algebra
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and you can rewrite this as
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the integral of udv
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u times v
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minus the integral of vdu.
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can all tell my u's from my v's?
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u's have the little tail on them.
00:07:04.520 --> 00:07:05.020
ok?
00:07:05.480 --> 00:07:08.440
so that is our integration by parts formula.
00:07:08.960 --> 00:07:12.080
so what'll happen is weve seen integrals now where you have a function
00:07:12.240 --> 00:07:13.360
and then you have
00:07:13.900 --> 00:07:15.680
a derivative inside the function
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and you say aha! chain rule.
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well sometimes you'll have an integral where you have a function and you have another function
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multiplied together but you cant use the chain rule.
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you cant use u substitution in other words.
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so these will often have been the product rule.
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so integration by parts
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works the product rule backwards.
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ok?
00:07:45.360 --> 00:07:49.540
this is a very powerful technique and you can often use this to solve lots of integrals.
00:07:49.540 --> 00:07:51.920
one of the problems i tend to run into is
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once we teach you this
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you tend to try to use this on everything.
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and you go into the exam and say lets do this with integration by parts
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doesnt work.
00:08:00.760 --> 00:08:02.880
i could do this one with integration by parts and that one doesnt work.
00:08:02.880 --> 00:08:04.960
there are other techniques out there okay?
00:08:05.340 --> 00:08:06.540
so dont just
00:08:06.820 --> 00:08:08.420
do integration by parts
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so far so good?
00:08:10.880 --> 00:08:14.120
alright so how do we do one of these? well lets take a nice simple example.
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suppose i ask you to do
00:08:16.380 --> 00:08:17.420
the integral of
00:08:18.160 --> 00:08:21.220
xcosx dx
00:08:23.920 --> 00:08:26.400
so you look at that and you say im gonna do u substitution.
00:08:26.860 --> 00:08:27.820
the problem is
00:08:28.320 --> 00:08:30.140
if i make this u
00:08:30.600 --> 00:08:32.760
thats not sinx thats not the derivative.
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if i make this u
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thats not 1 so thats not the derivative
00:08:36.780 --> 00:08:37.940
so i need something else.
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what i do is identify
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a u and a dv.
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and then im going to play with it a bit and use this formula.
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so i look at this and i say lets let u=x.
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dv=cosx dx.
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in other words inside the function i have something im going to differentiate
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and something im going to integrate.
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now i take the derivative of u
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i get just dx..1dx.
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and i take the antiderivative of dv remember thats really 1dv.
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and i get sinx
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dx.
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so now my formula says
00:09:34.760 --> 00:09:37.880
the integral of xcosx dx
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is u times v
00:09:42.000 --> 00:09:43.520
x times sinx
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minus..
00:09:45.940 --> 00:09:48.600
the integral of vdu
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which is sinx
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dx. that dx shouldnt have been there okay?
00:09:57.260 --> 00:10:00.780
and now the integral of sinx is easy thats just -cosx.
00:10:01.560 --> 00:10:05.780
so this is xsinx+cosx
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dont forget the plus c.
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okay? never forget that.
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thats it, thats the solution.
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yes?
00:10:17.840 --> 00:10:18.480
of course
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so
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i look at this integral and i say one part of this im going to take the derivative of
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and one part take the antiderivative of.
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so i say well lets let u=x
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ill tell you in a little why i chose that one.
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dv=cosx
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so im going to differentiate u and im going to get 1dx
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and you integrate cosx and i get sinx.
00:10:43.780 --> 00:10:48.980
and now i go to my formula and it says the integral of udv so this is u and this is dv
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will equal
00:10:51.460 --> 00:10:52.440
u times v
00:10:53.520 --> 00:10:54.960
times the integral of
00:10:56.000 --> 00:10:57.740
v du.
00:11:03.060 --> 00:11:03.560
ok?
00:11:04.640 --> 00:11:05.920
so then i plug in
00:11:06.140 --> 00:11:08.440
i say well u times v is x times sinx
00:11:09.280 --> 00:11:09.780
and
00:11:10.240 --> 00:11:11.900
v du is sinxdx.
00:11:12.440 --> 00:11:14.440
i integrate that and i get cosx.
00:11:16.160 --> 00:11:19.380
how do i know this works? well lets take the derivative of this just to prove it.
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lets do the check.
00:11:25.260 --> 00:11:32.020
i have xsinx+cosx+ c and im going to take the derivative of that.
00:11:35.080 --> 00:11:38.500
so lets do the product rule because remember i dont want to end up with xcosx.
00:11:39.080 --> 00:11:42.580
so if you said well what if its just xsinx? thats kind of what youre doing.
00:11:43.040 --> 00:11:45.740
see the problem is if i take the derivative here
00:11:46.140 --> 00:11:48.940
i get xcosx thats what i wanted
00:11:49.740 --> 00:11:50.240
plus
00:11:50.920 --> 00:11:52.500
1 times sinx
00:11:53.220 --> 00:11:54.840
with that annoying product rule.
00:11:55.260 --> 00:11:57.820
but now if i take the derivative of cosx
00:11:58.960 --> 00:11:59.980
minus sinx
00:12:01.220 --> 00:12:04.260
and i get what i wanted cause thats inside the integral.
00:12:04.660 --> 00:12:06.440
so thats how i know i got the right answer.
00:12:07.560 --> 00:12:08.120
i got this
00:12:08.500 --> 00:12:12.260
and i checked it by taking the derivative and i say aha i got xcosx.
00:12:13.900 --> 00:12:16.800
okay? we'll do a bunch of these dont worry
00:12:18.420 --> 00:12:20.900
so what the integration by parts does
00:12:21.460 --> 00:12:22.580
is it figures out
00:12:23.000 --> 00:12:24.980
the extra piece from the product rule
00:12:24.980 --> 00:12:26.580
and helps you subtract it off.
00:12:27.040 --> 00:12:29.200
okay cause like i said i looked at
00:12:29.500 --> 00:12:30.260
xcosx
00:12:30.260 --> 00:12:32.260
i said what if the antiderivative is just xsinx?
00:12:32.700 --> 00:12:33.820
it'll all be okay
00:12:34.380 --> 00:12:38.220
except when i take the derivative of xsinx i get this extra sinx piece.
00:12:38.920 --> 00:12:44.600
so if i have a cosx here the derivative of that is -sinx then they subtract and i get what i want.
00:12:47.220 --> 00:12:48.500
lets do another one
00:12:49.400 --> 00:12:50.360
so far so good?
00:12:51.780 --> 00:12:53.620
im going to erase it in a minute so.
00:12:54.460 --> 00:12:56.340
write it down, take a picture
00:12:57.020 --> 00:12:58.800
keep texting its all the same to me.
00:13:11.700 --> 00:13:12.660
so far so good?
00:13:13.620 --> 00:13:14.260
alright.
00:13:16.880 --> 00:13:18.800
do another one very similar.
00:13:29.080 --> 00:13:31.080
suppose i was going to do xe^x.
00:13:42.900 --> 00:13:45.720
again, i look at u substitution i say thats not going to work.
00:13:47.980 --> 00:13:48.540
because
00:13:49.100 --> 00:13:51.600
i dont have the function and its derivative inside.
00:13:52.060 --> 00:13:53.740
i just have 2 different functions in there.
00:13:58.720 --> 00:14:00.160
so if i let u=x
00:14:01.700 --> 00:14:02.200
du
00:14:03.640 --> 00:14:04.140
is dx.
00:14:04.580 --> 00:14:06.120
cause thats the derivative of u.
00:14:06.440 --> 00:14:09.420
alright the derivative of u, u is x, the derivative of u is dx.
00:14:11.740 --> 00:14:13.140
dv would be
00:14:13.760 --> 00:14:15.100
e^xdx.
00:14:15.460 --> 00:14:17.000
and whats the integral of e^x?
00:14:17.260 --> 00:14:17.920
e^x.
00:14:28.520 --> 00:14:30.040
okay so use the formula.
00:14:30.460 --> 00:14:31.480
formula says
00:14:31.880 --> 00:14:33.720
that the integral of udv
00:14:34.260 --> 00:14:38.380
is u times v minus the integral of vdu
00:14:38.380 --> 00:14:40.720
professor bearnhart calls this the voodoo
00:14:41.520 --> 00:14:42.160
formula.
00:14:43.660 --> 00:14:45.460
which the first time somebody said that to me
00:14:45.800 --> 00:14:47.000
i sort of said huh?
00:14:48.240 --> 00:14:49.560
he said just do voodoo right?
00:14:49.560 --> 00:14:50.840
i said im not quite sure what that is
00:14:50.840 --> 00:14:53.220
then i figured it out and i said yeah sure.
00:14:58.480 --> 00:14:59.380
this says
00:15:00.020 --> 00:15:01.900
u=x, du=dx.
00:15:02.660 --> 00:15:04.640
this dv=e^xdx.
00:15:05.480 --> 00:15:07.020
v is e^x.
00:15:11.120 --> 00:15:12.720
my handwriting is insulting.
00:15:15.340 --> 00:15:16.180
very sad.
00:15:18.120 --> 00:15:19.520
we dont love my handwriting?
00:15:21.180 --> 00:15:22.980
its not that bad for a math geek right?
00:15:23.060 --> 00:15:24.260
gets the job done.
00:15:25.260 --> 00:15:25.900
alright.
00:15:26.560 --> 00:15:27.060
so
00:15:27.280 --> 00:15:28.640
im plugging in parts
00:15:29.980 --> 00:15:30.720
u is x
00:15:31.140 --> 00:15:32.300
dv is e^x right?
00:15:32.440 --> 00:15:35.040
so the integral of xe^xdx
00:15:36.920 --> 00:15:39.220
equals u times v so that would be
00:15:39.540 --> 00:15:40.740
xe^x
00:15:41.640 --> 00:15:42.200
minus..
00:15:42.920 --> 00:15:43.960
the integral of
00:15:44.440 --> 00:15:49.720
vdu which is e^xdx.
00:15:52.960 --> 00:15:55.520
alright now whats the integral of e^x?
00:15:56.300 --> 00:15:56.800
e^x.
00:15:57.600 --> 00:16:01.540
so this is xe^x-e^x
00:16:02.780 --> 00:16:03.280
plus c.
00:16:05.520 --> 00:16:07.680
im the one allergic to chalk dust
00:16:07.680 --> 00:16:09.860
i should be coughing and sneezing like crazy.
00:16:13.600 --> 00:16:14.480
got the idea?
00:16:14.880 --> 00:16:17.480
lets have you guys do another one but before we do that
00:16:17.480 --> 00:16:21.040
i just want to show you a very simple rule that will be very useful for
00:16:21.340 --> 00:16:22.460
everything we do
00:16:22.540 --> 00:16:23.580
going forward.
00:16:24.160 --> 00:16:25.460
k if you have
00:16:26.540 --> 00:16:28.240
an integral of cosx
00:16:28.400 --> 00:16:30.480
right? we know thats just sinx.
00:16:30.480 --> 00:16:31.740
if you have a constant
00:16:35.100 --> 00:16:37.240
thats going to equal sin of kx
00:16:38.840 --> 00:16:40.140
divided by k.
00:16:41.120 --> 00:16:45.000
youd get that if you used u substitution but you dont want to use u substitution on simple ones like this.
00:16:45.000 --> 00:16:46.180
its going to drive you nuts.
00:16:46.180 --> 00:16:48.500
so when you take the derivative of this right?
00:16:48.500 --> 00:16:51.940
you would multiply by k so when you do the antiderivative you divide by k.
00:16:52.480 --> 00:16:53.920
so for example if i had
00:16:54.380 --> 00:16:56.500
integral of sin(3x)
00:16:58.860 --> 00:17:01.060
it would be -cos(3x)
00:17:02.420 --> 00:17:03.220
divided by 3.
00:17:04.700 --> 00:17:05.200
k?
00:17:05.720 --> 00:17:08.440
if thats true for cosine its true for sin.
00:17:21.280 --> 00:17:23.360
its uhh..im going to erase this
00:17:23.600 --> 00:17:24.660
useful for e.
00:17:27.500 --> 00:17:29.180
by the way secant squared
00:17:29.900 --> 00:17:31.040
no sec^2
00:17:31.040 --> 00:17:33.000
those ones we dont really like
00:17:33.140 --> 00:17:34.100
you could have
00:17:34.380 --> 00:17:35.280
e^kx
00:17:36.600 --> 00:17:37.100
dx
00:17:38.100 --> 00:17:41.860
and that becomes e^kx/k
00:17:42.500 --> 00:17:43.780
plus the constant.
00:17:44.960 --> 00:17:46.880
and one more thats very handy
00:17:47.760 --> 00:17:52.360
you have the integral of dx/ax+b
00:17:55.860 --> 00:17:56.980
thats going to be
00:17:57.920 --> 00:18:00.020
ln(ax+b)
00:18:01.480 --> 00:18:02.400
divded by a.
00:18:03.460 --> 00:18:06.500
so its all sort of the opposite of derivatives
00:18:06.500 --> 00:18:08.320
which makes sense cause these are antiderivatives.
00:18:13.240 --> 00:18:13.740
plus c.
00:18:19.120 --> 00:18:22.100
told you when i was done with my exams i just put plus c next to everything.
00:18:22.940 --> 00:18:24.460
just go back and check.
00:18:29.920 --> 00:18:32.440
so you should memorize those because theyre very handy.
00:18:32.780 --> 00:18:35.800
so we could give you something slightly harder and youd be able to do it.
00:18:37.800 --> 00:18:40.580
alright lets have you practice an integration by parts problem.
00:18:49.000 --> 00:18:52.240
and the hardest part that people have with integration by parts
00:18:52.240 --> 00:18:54.360
other than just executing correctly
00:18:54.800 --> 00:18:56.160
if the minus signs..
00:18:56.740 --> 00:18:58.020
and the fractions.
00:18:59.180 --> 00:19:00.540
tend to mess that up.
00:19:00.820 --> 00:19:02.220
so you know what youre doing
00:19:02.660 --> 00:19:05.000
but you have a little trouble with the minuses
00:19:05.120 --> 00:19:07.040
and fractions cause youll see
00:19:07.040 --> 00:19:09.220
sometimes you need to do integration by parts
00:19:09.540 --> 00:19:10.580
more than once.
00:19:13.360 --> 00:19:13.860
now
00:19:14.720 --> 00:19:16.880
take one very similar to the one we just did
00:19:17.740 --> 00:19:19.100
lets do the integral
00:19:19.920 --> 00:19:25.680
of xsin(3x)dx.
00:19:26.580 --> 00:19:28.700
why doesnt everyone take 2 minutes and see if you can solve it.
00:19:30.140 --> 00:19:31.640
you need a functions derivative
00:19:31.840 --> 00:19:33.360
as a general rule
00:19:33.360 --> 00:19:36.320
if you have x to a power youre going to let that be the function.
00:19:37.100 --> 00:19:38.940
and the other thing would be
00:19:39.300 --> 00:19:40.580
the derivative. so
00:19:40.580 --> 00:19:43.140
in other words you let u=x to the something
00:19:43.140 --> 00:19:45.140
and dv will be the other piece.
00:19:45.520 --> 00:19:47.900
so itd be like u=the x
00:19:48.860 --> 00:19:49.540
dv
00:19:50.320 --> 00:19:52.200
would be sin(3x)..
00:19:53.160 --> 00:19:53.660
dx
00:19:54.480 --> 00:19:55.680
du..take this one
00:19:56.920 --> 00:19:58.040
integrate that.
00:20:00.360 --> 00:20:02.440
the derivative of x is 1dx.
00:20:04.400 --> 00:20:06.240
antiderivative of sin(3x)
00:20:07.340 --> 00:20:09.240
is -cos(3x).
00:20:10.540 --> 00:20:12.780
over 3, just taught you guys that.
00:20:13.700 --> 00:20:14.200
ok?
00:20:15.060 --> 00:20:18.020
cause you anti-differentiate, youre integrating, okay?
00:20:20.620 --> 00:20:22.300
now we plug this in the formula.
00:20:22.600 --> 00:20:26.060
formula is udv..integral of udv
00:20:27.340 --> 00:20:31.020
equals u times v minus the integral of vdu.
00:20:31.600 --> 00:20:34.360
so the integral of xsin(3x)
00:20:37.460 --> 00:20:39.080
dx is..
00:20:40.240 --> 00:20:41.440
u times v
00:20:42.000 --> 00:20:43.200
so this times this
00:20:44.440 --> 00:20:50.180
-xcos(3x)/3.
00:20:52.100 --> 00:20:52.740
minus...
00:20:52.740 --> 00:20:55.960
the integral of vdu but we have a minus so that equals plus
00:20:56.760 --> 00:21:02.160
the integral of cos(3x)/3dx
00:21:02.160 --> 00:21:04.680
remember i said watch your minus signs and watch your fractions.
00:21:10.120 --> 00:21:11.080
so far so good?
00:21:15.880 --> 00:21:17.600
now we just have to integrate this.
00:21:18.940 --> 00:21:19.580
well this
00:21:19.980 --> 00:21:20.620
is just..
00:21:21.740 --> 00:21:23.020
we still have cos3x
00:21:24.400 --> 00:21:28.080
over 3 is just 1/3 the integral of cos(3x)
00:21:29.220 --> 00:21:30.540
so thats 1/3
00:21:31.460 --> 00:21:32.600
times sin3x
00:21:34.700 --> 00:21:35.520
over 3.
00:21:37.060 --> 00:21:38.180
also known as
00:21:38.580 --> 00:21:39.620
sin3x
00:21:41.700 --> 00:21:42.200
over 9.
00:21:44.160 --> 00:21:46.080
so this becomes-i know plus c
00:21:46.480 --> 00:21:47.040
becomes
00:21:47.580 --> 00:21:52.820
-xcos3x/3
00:21:54.380 --> 00:21:54.880
plus
00:21:56.180 --> 00:22:02.780
sin3x/9+c okay? so again
00:22:03.680 --> 00:22:05.140
you do, udv
00:22:05.380 --> 00:22:06.640
so u is x
00:22:07.100 --> 00:22:08.540
dv is sin3x.
00:22:09.220 --> 00:22:10.500
derivative of x is 1
00:22:11.100 --> 00:22:14.820
the antiderivative of sin is -cos3x/3.
00:22:16.060 --> 00:22:17.380
now you go to your formula.
00:22:18.540 --> 00:22:20.480
you have..minus..
00:22:20.980 --> 00:22:24.320
xcos3x/3, thats the first part.
00:22:25.700 --> 00:22:27.380
minus minus becomes plus
00:22:27.840 --> 00:22:29.520
the integral of cos3x/3.
00:22:30.540 --> 00:22:31.500
this integral
00:22:32.340 --> 00:22:34.440
well its 1/3cos3x
00:22:34.880 --> 00:22:36.960
so thats going to be 1/3sin3x/3
00:22:37.560 --> 00:22:38.680
1/9sin3x.
00:22:39.420 --> 00:22:41.000
k and you just put the whole thing together.
00:22:42.100 --> 00:22:44.660
so whatll happen with integration by parts is you get these chains
00:22:45.260 --> 00:22:47.260
of functions because as i said
00:22:47.480 --> 00:22:49.340
you need the other terms
00:22:49.740 --> 00:22:51.740
to get rid of the problem terms
00:22:51.740 --> 00:22:52.780
from the product rule.
00:22:53.180 --> 00:22:59.020
because you look at this and you say i want this to just be -xcos3x but its not that simple.
00:22:59.580 --> 00:23:01.340
if we differentiated this
00:23:01.440 --> 00:23:03.040
you have a leftover term
00:23:03.040 --> 00:23:05.140
and this helps you get rid of the leftover term.
00:23:06.700 --> 00:23:07.200
k?
00:23:09.180 --> 00:23:10.480
anybody get this one right?
00:23:11.060 --> 00:23:11.860
some of you?
00:23:12.540 --> 00:23:13.040
yay!
00:23:13.040 --> 00:23:14.380
alright lets do another one.
00:23:15.780 --> 00:23:17.300
gotta get good at this.
00:23:18.180 --> 00:23:18.900
how about...
00:23:19.740 --> 00:23:23.120
the integral of x^2e^2x
00:23:24.160 --> 00:23:25.440
dx. now ill tell you
00:23:26.180 --> 00:23:27.200
that x^2
00:23:27.200 --> 00:23:30.640
youre going to have to do integration by parts twice.
00:23:31.580 --> 00:23:32.080
k?
00:23:32.460 --> 00:23:34.100
so let u=x^2
00:23:34.100 --> 00:23:37.280
integration by parts, and then when youre done you have to do it a second time.
00:23:54.340 --> 00:23:54.840
so
00:23:55.400 --> 00:23:57.560
lets let u=x^2
00:24:01.120 --> 00:24:04.040
and dv=e^2x
00:24:05.160 --> 00:24:05.660
dx.
00:24:07.460 --> 00:24:07.960
ok?
00:24:09.080 --> 00:24:09.760
then du
00:24:10.340 --> 00:24:11.060
will equal
00:24:12.580 --> 00:24:13.840
2xdx.
00:24:16.220 --> 00:24:20.380
v will equal e^2x/2.
00:24:24.360 --> 00:24:26.680
in general when you have x to a power
00:24:27.260 --> 00:24:30.880
youll usually have e, sin or cos as the other term, usually.
00:24:31.320 --> 00:24:34.240
then x to the power will be your u term.
00:24:34.980 --> 00:24:37.620
the main exception will be natural log.
00:24:38.020 --> 00:24:41.380
if you have lnx thats always going to be your u term.
00:24:41.580 --> 00:24:44.300
these are the types that you can count on.
00:24:46.280 --> 00:24:47.280
okay so
00:24:48.040 --> 00:24:50.460
now we have the integral of x^2e^2x
00:24:51.540 --> 00:24:52.040
is..
00:24:52.820 --> 00:24:53.940
u times v.
00:24:54.780 --> 00:24:57.400
so x^2e^2x/2
00:24:58.260 --> 00:24:59.020
minus..
00:24:59.460 --> 00:25:00.820
the integral of vdu.
00:25:00.980 --> 00:25:02.580
which is the integral of
00:25:03.100 --> 00:25:04.620
e^2x/2
00:25:06.400 --> 00:25:07.560
times 2x
00:25:09.000 --> 00:25:10.600
dx and those 2's cancel.
00:25:11.520 --> 00:25:12.600
so you have
00:25:13.180 --> 00:25:17.000
x^2e^2x/2
00:25:17.560 --> 00:25:22.860
minus the integral of xe^2xdx.
00:25:25.960 --> 00:25:28.520
we just did the integral of xe^x right?
00:25:29.720 --> 00:25:32.200
when it says do that just do integration by parts.
00:25:38.840 --> 00:25:39.340
now
00:25:41.320 --> 00:25:43.760
write u=x
00:25:44.820 --> 00:25:46.340
du will be dx.
00:25:47.940 --> 00:25:50.820
dv is e^2xdx.
00:25:52.600 --> 00:25:56.840
v is e^2x/2.
00:26:07.360 --> 00:26:08.640
so this now becomes
00:26:10.860 --> 00:26:15.020
x^2e^2x/2
00:26:15.660 --> 00:26:16.220
minus..
00:26:17.180 --> 00:26:18.900
be very careful about minus signs.
00:26:18.900 --> 00:26:20.460
very easy to mess them up.
00:26:21.940 --> 00:26:28.180
u times v is xe^2x/2.
00:26:29.920 --> 00:26:30.840
minus
00:26:32.600 --> 00:26:33.400
integral of
00:26:33.940 --> 00:26:37.980
e^2x/2 dx.
00:26:38.360 --> 00:26:40.600
and you distribute the minus sign
00:26:40.920 --> 00:26:45.220
you get x^2e^2x/2
00:26:45.760 --> 00:26:50.400
minus xe^2x/2.
00:26:51.880 --> 00:26:52.380
plus
00:26:53.340 --> 00:26:56.640
1/2 the integral of e^2x
00:26:58.940 --> 00:26:59.440
dx
00:27:02.620 --> 00:27:05.040
one more round now you can do integration by parts.
00:27:05.040 --> 00:27:06.620
thats just a basic integral.
00:27:07.640 --> 00:27:08.680
this becomes..
00:27:09.660 --> 00:27:12.920
x^2e^2x/2
00:27:14.880 --> 00:27:15.440
minus..
00:27:16.240 --> 00:27:19.640
xe^2x/2.
00:27:20.540 --> 00:27:23.660
plus e^2x so lets see
00:27:23.960 --> 00:27:24.840
this is over 2
00:27:25.140 --> 00:27:26.180
and thats a half
00:27:26.720 --> 00:27:27.500
so over 4.
00:27:28.720 --> 00:27:29.840
plus a constant.
00:27:32.860 --> 00:27:35.640
and if you differentiate that youll see how the product rule
00:27:36.180 --> 00:27:36.680
works.
00:27:39.000 --> 00:27:41.540
so you take the derivative of this
00:27:42.700 --> 00:27:44.480
you have a leftover term from the product rule.
00:27:44.480 --> 00:27:48.880
just gets rid of it but wait you take the derivative of this you have a leftover term from the product rule
00:27:48.880 --> 00:27:50.260
and this gets rid of it.
00:27:54.300 --> 00:27:55.580
gotta do some more.
00:27:55.860 --> 00:27:57.240
get you guys good at these.
00:28:00.820 --> 00:28:02.180
how much fun is this?
00:28:02.900 --> 00:28:03.400
no?
00:28:04.440 --> 00:28:06.020
youll get good at these youll see.
00:28:07.900 --> 00:28:09.740
e^x take you all night.
00:28:13.840 --> 00:28:15.200
whered you get lost?
00:28:16.500 --> 00:28:17.000
here?
00:28:17.860 --> 00:28:19.640
minus xe^2x
00:28:19.640 --> 00:28:23.060
minus minus becomes plus i just took the 2 out and thats a half
00:28:27.620 --> 00:28:28.820
and this is a minus
00:28:29.060 --> 00:28:30.980
this is a minus so thats a plus
00:28:31.420 --> 00:28:33.960
and this 2 becomes a half.
00:28:43.500 --> 00:28:45.980
you have x^2e^2/2 minus this one
00:28:46.120 --> 00:28:48.760
minus minus becomes a plus the last one.
00:28:49.380 --> 00:28:49.880
k?
00:28:50.900 --> 00:28:54.080
e^2x/2 is the same as a half e^2x.
00:28:54.540 --> 00:28:56.140
thats what it means divided by 2 right?
00:28:57.780 --> 00:28:59.340
k youre not happy.
00:28:59.580 --> 00:29:00.940
lets do another one.
00:29:03.480 --> 00:29:07.000
by the way you could have like x^3, x^4, x^5 but we wouldnt do that of course
00:29:07.500 --> 00:29:08.460
theres a trick
00:29:08.740 --> 00:29:10.580
if you have high powers of x.
00:29:12.880 --> 00:29:15.280
or a technique its not really a trick
00:29:15.540 --> 00:29:16.340
a shortcut.
00:29:29.580 --> 00:29:31.860
okay, gotta do that one by parts.
00:29:33.920 --> 00:29:34.640
its bit of pain
00:29:35.160 --> 00:29:36.520
but we'll get there.
00:29:39.020 --> 00:29:40.780
we have 2 other types to learn today.
00:29:46.280 --> 00:29:48.680
remember this isnt u substitution.
00:29:48.680 --> 00:29:51.600
dont confuse the u from the one with the u from the other.
00:29:51.600 --> 00:29:53.420
just kept it the same letter.
00:30:06.240 --> 00:30:08.880
okay lets let u=x^2.
00:30:10.300 --> 00:30:13.380
dv will be cos4x.
00:30:15.400 --> 00:30:15.900
dx
00:30:16.600 --> 00:30:18.200
if the u=x^2
00:30:18.200 --> 00:30:20.720
cause when i differentiate x^2 i get 2x.
00:30:20.720 --> 00:30:22.540
when i differentiate again i get 2.
00:30:22.540 --> 00:30:24.780
my problem gets better each time.
00:30:24.780 --> 00:30:26.660
if i differentiate cos i just get sin
00:30:26.660 --> 00:30:28.720
if i differentiate sin i just get back to cos
00:30:28.720 --> 00:30:30.400
so im not really getting anywhere.
00:30:31.740 --> 00:30:34.760
so thats why you let u be the x^2 term.
00:30:35.760 --> 00:30:37.920
if you do it the other way youll watch its just a mess.
00:30:38.960 --> 00:30:40.840
alright now i let du
00:30:42.220 --> 00:30:44.220
=2x dx.
00:30:46.680 --> 00:30:50.580
and v will be sin4x/4.
00:30:51.320 --> 00:30:53.640
its always divided by this number.
00:30:55.680 --> 00:31:02.640
so now this integral x^2cos4xdx becomes..
00:31:03.600 --> 00:31:05.680
the integral of u times v
00:31:06.500 --> 00:31:13.060
which is x^2sin4x/4.
00:31:14.100 --> 00:31:16.340
minus the integral of vdu.
00:31:16.540 --> 00:31:17.100
minus..
00:31:18.320 --> 00:31:19.120
integral of
00:31:19.680 --> 00:31:22.940
2xsin4x/4
00:31:26.800 --> 00:31:27.300
dx.
00:31:27.660 --> 00:31:29.260
and the 2 and the 4 cancel
00:31:29.980 --> 00:31:31.740
so i get this is x^2
00:31:33.140 --> 00:31:34.220
sin4x
00:31:36.180 --> 00:31:36.680
over 4
00:31:37.420 --> 00:31:37.920
minus
00:31:38.880 --> 00:31:39.380
x-oops
00:31:42.900 --> 00:31:44.000
2/4 is a half.
00:31:44.000 --> 00:31:45.880
so take that out-minus a half
00:31:46.660 --> 00:31:48.400
integral of xsin4x
00:31:51.000 --> 00:31:51.500
dx.
00:31:52.780 --> 00:31:56.860
so ive improved my integration by parts by going from x^2 to x.
00:31:57.500 --> 00:31:59.100
so now i just do it again.
00:32:06.080 --> 00:32:06.580
so i let
00:32:07.620 --> 00:32:08.840
u=x
00:32:10.700 --> 00:32:14.500
dv..sin4xdx.
00:32:17.820 --> 00:32:19.220
then du is dx.
00:32:20.420 --> 00:32:23.920
v is -cos4x
00:32:29.080 --> 00:32:29.580
dx.
00:32:32.580 --> 00:32:33.960
no dx sorry.
00:32:34.440 --> 00:32:35.760
-cos4x.
00:32:37.640 --> 00:32:40.960
so this integral-remember our original integral was x^2..
00:32:42.020 --> 00:32:45.780
cos4xdx
00:32:46.620 --> 00:32:47.500
now equals
00:32:48.460 --> 00:32:52.080
x^2sin4x/4
00:32:53.860 --> 00:32:56.500
minus a half of..now this integral
00:32:57.260 --> 00:33:02.560
becomes -xcos4x
00:33:04.360 --> 00:33:05.660
minus minus
00:33:07.080 --> 00:33:07.880
minus minus
00:33:09.420 --> 00:33:10.220
integral of
00:33:11.280 --> 00:33:13.360
cos4x
00:33:15.460 --> 00:33:15.960
dx.
00:33:17.180 --> 00:33:17.680
yes?
00:33:21.200 --> 00:33:22.720
oh i forgot over 4 sorry
00:33:26.000 --> 00:33:26.720
thank you.
00:33:29.860 --> 00:33:31.900
be careful with the minus signs okay?
00:33:32.120 --> 00:33:33.480
you have minus a half
00:33:34.020 --> 00:33:35.720
times this integral.
00:33:36.200 --> 00:33:39.080
which is -xcos4x/4
00:33:39.980 --> 00:33:42.220
minus..minus..
00:33:42.680 --> 00:33:43.880
that becomes plus
00:33:44.460 --> 00:33:46.460
integral of cos4x/4
00:33:46.740 --> 00:33:47.240
dx.
00:33:49.560 --> 00:33:51.460
distribute. watch your halves.
00:33:51.460 --> 00:33:52.900
watch your minus signs.
00:33:53.920 --> 00:33:59.380
you get x^2sin4x/4
00:34:00.180 --> 00:34:01.220
plus an eighth.
00:34:02.640 --> 00:34:05.520
of xcos4x
00:34:06.880 --> 00:34:07.920
minus an eighth
00:34:08.900 --> 00:34:09.940
the integral of
00:34:10.680 --> 00:34:13.920
cos4xdx.
00:34:15.580 --> 00:34:17.660
and that is sin4x/4.
00:34:18.260 --> 00:34:18.820
so we get
00:34:19.700 --> 00:34:22.640
x^2sin4x
00:34:25.200 --> 00:34:25.700
over 4
00:34:27.460 --> 00:34:32.100
plus 1/8xcos4x.
00:34:32.920 --> 00:34:36.740
-1/8sin4x/4
00:34:36.740 --> 00:34:38.600
it becomes sin4x/32
00:34:40.100 --> 00:34:43.240
plus c so see how it goes 4, 8, 32?
00:34:43.680 --> 00:34:44.180
right?
00:34:46.240 --> 00:34:48.100
youll see a lot of that kind of stuff.
00:34:49.520 --> 00:34:51.200
that comes from all of the
00:34:51.480 --> 00:34:53.800
differentiating in product rule.
00:34:55.440 --> 00:34:56.400
that was messy.
00:34:58.400 --> 00:34:59.920
thats all one problem.
00:35:01.040 --> 00:35:02.960
grading these is really fun.
00:35:04.120 --> 00:35:04.760
sarcasm.
00:35:05.780 --> 00:35:06.280
yes?
00:35:07.160 --> 00:35:09.160
i hope you get partial credit.
00:35:11.040 --> 00:35:11.840
not up to me.
00:35:13.860 --> 00:35:16.820
yes of course youll get some partial credit.
00:35:18.320 --> 00:35:20.900
i dont tend to assign anything harder than this
00:35:20.900 --> 00:35:22.780
because like i said its not fun to grade
00:35:23.100 --> 00:35:28.060
and also were really just testing peoples algebra skills at some point youre not testing calculus skills.
00:35:29.560 --> 00:35:32.340
alright lets do an easier type
00:35:32.340 --> 00:35:35.580
but another one when i say easier i mean less algebraic.
00:35:37.840 --> 00:35:40.120
you guys wanted to take even more calculus.
00:35:41.940 --> 00:35:44.120
no want here. have to.
00:35:47.560 --> 00:35:48.060
must.
00:35:49.040 --> 00:35:49.540
right?
00:35:50.140 --> 00:35:52.960
youre here because you must be here not because you wish to be here.
00:35:54.000 --> 00:35:55.520
with a few exceptions.
00:36:04.180 --> 00:36:07.460
we know that the derivative of natural log is 1/x.
00:36:07.800 --> 00:36:10.360
and the integral of 1/x is natural log.
00:36:10.360 --> 00:36:12.640
but what do you think the integral of natural log is?
00:36:17.260 --> 00:36:19.900
we kind of want that to be 1/x but its not.
00:36:19.900 --> 00:36:20.860
its the other way around.
00:36:22.640 --> 00:36:24.820
we do that with integration by parts.
00:36:26.260 --> 00:36:26.840
let u
00:36:28.380 --> 00:36:29.460
=lnx
00:36:31.820 --> 00:36:33.960
dv, well whats dv equal to?
00:36:35.900 --> 00:36:36.400
dv
00:36:36.840 --> 00:36:38.200
is just going to be dx.
00:36:38.200 --> 00:36:40.740
its going to be 1, remember this is a 1 in here.
00:36:41.340 --> 00:36:41.840
so dv
00:36:43.340 --> 00:36:43.840
is dx.
00:36:44.100 --> 00:36:45.400
so when you differentiate this
00:36:45.960 --> 00:36:48.080
you get 1/xdx.
00:36:48.380 --> 00:36:51.300
when you integrate this you get x.
00:36:59.320 --> 00:37:01.300
so now i go to the formula
00:37:03.220 --> 00:37:03.940
and this is
00:37:04.900 --> 00:37:08.340
u times v which is xlnx
00:37:09.040 --> 00:37:09.540
minus
00:37:10.720 --> 00:37:12.860
the integral of v times du.
00:37:14.400 --> 00:37:15.520
whats v times du?
00:37:17.520 --> 00:37:18.020
x
00:37:19.260 --> 00:37:19.980
times 1/x.
00:37:26.860 --> 00:37:28.460
and x and the 1/x cancel.
00:37:28.600 --> 00:37:29.560
so this equals
00:37:29.940 --> 00:37:31.600
x times lnx
00:37:32.200 --> 00:37:33.880
times the integral of dx.
00:37:36.980 --> 00:37:38.820
the integral of dx is just x.
00:37:39.260 --> 00:37:41.200
the derivative of x is 1 remember this is
00:37:41.680 --> 00:37:43.200
its saying this is 1dx.
00:37:45.440 --> 00:37:46.900
so this is x..
00:37:47.520 --> 00:37:48.020
lnx
00:37:49.140 --> 00:37:49.640
minus x
00:37:50.140 --> 00:37:51.660
you may want to memorize that.
00:37:52.020 --> 00:37:53.060
i memorized it.
00:37:53.480 --> 00:37:54.760
i recommend you do.
00:38:03.700 --> 00:38:05.640
so the integral of log of x
00:38:06.720 --> 00:38:07.220
dx
00:38:08.100 --> 00:38:12.680
equals xlnx-x
00:38:15.500 --> 00:38:18.300
a handy one just to put in your memory bank.
00:38:30.320 --> 00:38:31.280
so far so good?
00:38:38.200 --> 00:38:39.800
so far so good, alright.
00:38:39.800 --> 00:38:42.820
one last one for today cause i know how much you love this.
00:38:47.040 --> 00:38:49.840
this is one of the more interesting types.
00:38:51.420 --> 00:38:53.180
we did the x times e^x
00:38:53.300 --> 00:38:54.900
the x times sinx
00:38:54.900 --> 00:38:56.820
the x^2 times e^x and all that stuff
00:38:56.900 --> 00:38:58.920
right? now..
00:38:59.600 --> 00:39:01.200
slightly different one.
00:39:02.120 --> 00:39:04.220
what if i want to do the integral of..
00:39:05.280 --> 00:39:08.120
e^xcosx
00:39:08.980 --> 00:39:09.480
dx
00:39:10.900 --> 00:39:13.900
these things show up a lot in electrical engineering.
00:39:14.420 --> 00:39:15.620
and other places.
00:39:20.740 --> 00:39:22.020
show up in physics.
00:39:22.880 --> 00:39:26.280
lots of things that go like this you get the sines and cosines.
00:39:26.680 --> 00:39:27.880
and if they dampen
00:39:28.240 --> 00:39:29.040
so if they go
00:39:30.440 --> 00:39:31.920
then you get an e term.
00:39:32.640 --> 00:39:33.200
and so on
00:39:33.440 --> 00:39:34.660
so theres lots of phenomena
00:39:34.660 --> 00:39:35.820
that behave like these
00:39:35.940 --> 00:39:37.540
so you have to learn how to integrate them.
00:39:38.480 --> 00:39:40.680
theyre not the only ones, just some of them
00:39:41.020 --> 00:39:41.660
alright.
00:39:43.000 --> 00:39:45.420
lets do this one. lets let-well
00:39:45.940 --> 00:39:48.400
it doesnt really matter which one i pick see the problem is
00:39:49.060 --> 00:39:50.900
the derivative of e^x is e^x
00:39:50.900 --> 00:39:52.020
the integral of e^x is e^x
00:39:52.020 --> 00:39:53.380
so i dont see me getting anywhere.
00:39:53.380 --> 00:39:58.260
but when i do sin and cosine it just kind of floats back and forth so how am i going to do this one? well lets see
00:39:58.760 --> 00:40:00.480
lets let u=e^x.
00:40:00.940 --> 00:40:06.720
dv=cosxdx.
00:40:09.600 --> 00:40:12.420
du is e^xdx.
00:40:13.880 --> 00:40:17.380
v is sinx.
00:40:25.840 --> 00:40:26.340
so now
00:40:26.420 --> 00:40:27.460
the integral of
00:40:27.860 --> 00:40:29.540
e^xcosx
00:40:30.560 --> 00:40:31.060
dx
00:40:32.260 --> 00:40:34.980
equals u times v
00:40:35.740 --> 00:40:38.200
which is e^xsinx
00:40:38.800 --> 00:40:39.300
minus
00:40:40.060 --> 00:40:41.420
the integral of vdu.
00:40:42.020 --> 00:40:43.620
e^xsinx
00:40:46.300 --> 00:40:46.800
dx.
00:40:48.660 --> 00:40:52.940
i havent really made my life easier have i? i just turned e^xcosx into e^xsinx.
00:40:55.800 --> 00:40:57.000
but hang in there.
00:40:57.940 --> 00:40:58.580
alright?
00:40:59.220 --> 00:41:00.760
so i have to do this integral now
00:41:02.500 --> 00:41:06.040
so lets let u=e to the-nah ill put it over to the side.
00:41:11.020 --> 00:41:12.760
u=e^x.
00:41:14.940 --> 00:41:18.060
dv equals sinx dx.
00:41:20.660 --> 00:41:22.700
du will equal e^xdx.
00:41:23.880 --> 00:41:25.920
and v will equal -cosx.
00:41:29.760 --> 00:41:32.160
k be careful watch your minus signs.
00:41:32.520 --> 00:41:34.140
so im going to take this integral
00:41:34.840 --> 00:41:37.320
my original integral was e^xcosxdx.
00:41:38.680 --> 00:41:41.300
and that equals e^xsinx
00:41:43.820 --> 00:41:45.660
minus, now i use the formula
00:41:47.260 --> 00:41:49.920
minus e^xcosx.
00:41:51.040 --> 00:41:52.520
minus minus is plus.
00:41:54.360 --> 00:41:56.820
integral of e^xcosx
00:41:58.000 --> 00:41:58.500
dx
00:42:00.000 --> 00:42:00.700
which is
00:42:06.140 --> 00:42:06.940
this equals
00:42:07.160 --> 00:42:08.500
e^xsinx
00:42:10.740 --> 00:42:12.900
+ e^xcosx
00:42:14.800 --> 00:42:15.300
minus
00:42:15.920 --> 00:42:19.260
the integral of e^xcosxdx
00:42:20.420 --> 00:42:23.340
so now you say wait a second im back where i started.
00:42:23.920 --> 00:42:24.420
right?
00:42:25.180 --> 00:42:26.900
i had e^xcosx
00:42:27.200 --> 00:42:29.120
and it turned it into e^xsinx
00:42:29.440 --> 00:42:31.500
and now im back to e^xcosx.
00:42:31.500 --> 00:42:33.880
this is a waste of time youre just going around in circles.
00:42:34.360 --> 00:42:36.240
i dont understand, help!
00:42:37.880 --> 00:42:39.640
so then you go-well wait..
00:42:40.300 --> 00:42:42.880
this is a minus integral of e^xcosx.
00:42:42.880 --> 00:42:45.240
so why dont i just add the integral back to the other side?
00:42:45.680 --> 00:42:47.340
yes, you can do that.
00:42:47.340 --> 00:42:49.940
what do you mean you can do that, im going to do it
00:42:50.300 --> 00:42:50.800
watch.
00:42:51.060 --> 00:42:52.560
so now you get 2
00:42:52.820 --> 00:42:55.300
integral e^xcosx
00:42:56.220 --> 00:42:56.720
dx
00:43:02.020 --> 00:43:02.660
equals..
00:43:03.340 --> 00:43:04.540
e^xsinx
00:43:05.940 --> 00:43:07.760
+e^xcosx
00:43:11.020 --> 00:43:11.820
youre done.
00:43:12.840 --> 00:43:15.240
wait what do you mean im done? youre done.
00:43:16.000 --> 00:43:16.920
divide by 2
00:43:17.920 --> 00:43:20.060
e^xcosxdx
00:43:22.280 --> 00:43:23.440
is e^xsinx
00:43:25.140 --> 00:43:26.820
+e^xcosx
00:43:28.960 --> 00:43:29.460
over 2
00:43:29.900 --> 00:43:32.780
plus c-kind of seems like a cheat doesnt it?
00:43:32.780 --> 00:43:36.520
what it was you just put it back where you started and you ended up with the final answer.
00:43:36.520 --> 00:43:40.160
said hey i have an integral on the left and i had no integral on the right so i must have done something correct.
00:43:40.520 --> 00:43:42.280
so lets go through it again.
00:43:42.660 --> 00:43:44.220
so you let u=e^x.
00:43:44.500 --> 00:43:46.260
du is e^xdx.
00:43:46.920 --> 00:43:50.240
dv is cosx and v is sinx so you get this integral.
00:43:51.440 --> 00:43:52.480
then you repeat
00:43:52.940 --> 00:43:53.820
you get that.
00:43:54.700 --> 00:43:55.500
here except
00:43:55.920 --> 00:43:57.200
now you have a minus
00:43:57.200 --> 00:43:58.840
the integral of e^xcosx.
00:43:59.380 --> 00:44:02.900
so if i add it back to the other side i have e^xcosx
00:44:03.300 --> 00:44:05.860
plus another integral of e^xcosx
00:44:07.000 --> 00:44:09.640
which should be 2 integrals of e^xcosx.
00:44:10.080 --> 00:44:11.600
then i just divide by 2.
00:44:13.980 --> 00:44:15.540
took this integral right here..
00:44:17.620 --> 00:44:19.700
took this integral right here
00:44:20.620 --> 00:44:22.620
and added it to the other side.
00:44:23.920 --> 00:44:26.080
and yes youre allowed to do that.
00:44:27.600 --> 00:44:29.440
alright ill see you monday.