Integration by Substitution

Start   and you started or mostly finished or made some progress on the homework assignment which was officially due at the last class.
but because of the hurricane you could finish it sometime before you sleep today.
if youre having trouble getting into webassign you should let me know so we can work around it.
theres a few people that had issues where they tried to log in and instead of being asked for their blackboard password
0:33they asked for the webassign password.
according to webassign thats due to the fact that you dont take webassign's cookies.
so you have to turn the cookies on your browser.
or you can just sign in through webassign.
any questions or..? from the first class?
student: is there a way to see if you correctly submitted your homework?
1:00so on your homework if theres a little green check next to your question.. it submits as you go.
when you put in-when you see a red x then that means you submitted it but its wrong.
if you see a little green check then you have those points. so its not like do your homework, you work on it, whatever okay turn it in!
it gets turned in as you go.
so when i said that
1:31some people are working on it and have started it and some people have finished thats because i can see that theres some people that have, i dont know, a point a half that means they got a question right and one question wrong then right and thats all theyve done.
so they may very well have 19 points by the end of the day.
okay? other questions? yeah?
no the paper homework is due next week there is no paper homework due this week.
2:05no it says the 14th, the 15th, or the16th.
so i put the homework on the week that corresponds to the material thats on the homework.
so its not the date that its due its on the week that its assigned.
so for example homework 2 on webassign is up and its on this week because its due next week.
2:34and also there was some confusion i know i asked it before but sometimes people dont listen or it doesnt sink in that if you do your homework problems more than 48 hours before theyre due you get extra credit.
so its quite possible for some of you to get a point a half for a problem because-for a problem thats worth 1 point so theres a lot of people first homework assigment was worth 20 points but theres a lot of people that have more than 20 points
3:00because they did it early.
other issues questions anything?
alright-oh whatever, okay so lets do some math now.
we were talking about we talked about what the integral is how it represents an area and we were focusing on definite integrals.
3:33which is something like.. this which corresponds to this area.
we also use in the process a similar
4:01notation without putting little numbers here.
so this is a number.
putting little numbers here means a function so this would be some function well i dont need a constant there.
so this is a function.
even though the notation is almost the same this is a function and we saw last time by the fundamental theorem
4:31that the derivative is f.
this is an antiderivative.
so sometimes this is called an antiderivative.
and theres lots of them.
so for example
5:01this should be nothing new to anybody.
for example if i write the integral from 1 to 2 of x^2dx then this is 1/3x^3 evaluated from 1 to 2.
which is 8/3-1/3 which is 7/3.
5:34if i write this then this is 1/3x^3 and there are infinitely many of these. so i can add any constant here i like so we usually write +c.
this is the statement that the derivative of this is this.
no matter what c is.
6:00again, better not be anything new.
just reminding you of whats going on here.
a lot of students are sloppy and they leave off this dx.
this dx is very important because it tells you what youre integrating with respect to.
but it also is something thats useful in calculating the integral.
which is a lot of what im going to talk about today.
so let me write something that looks almost the same.
6:35this well actually let me just write that so can someone tell me what this is?
you cant do it?
i can do it.
lets take a poll.
theres an answer b) cant do it.
7:02you in the back did you have a statement?
c) zx^2 anyone else?
any other suggestions?
7:33okay so im going to ask a clicker question now, its that.
wasnt the one i planned to ask but i'm going to ask.
there are 5 answers there.
channel 41 as usual.
so in fact
8:00so lets just report 15% of people picked this.
i should be able to remember a smaller number. 8% think this.
uhh big number.. 28% think this.
6% and 41%
8:30so the 6% think this and 41% think this.
okay this ones wrong.
this ones wrong.
this ones wrong.
so its 1 of these 2.
can someone-so this ones almost right.
can someone tell me whats wrong with this one?
yes, i need a constant.
9:01so the right answer is E because the right answer is z^2... x im sorry plus a constant.
why is that?
this dz tells me that the thing thats changing is a z.
not an x.
so x is just a number.
so when i integrate 43dz
9:31the answer is 43z plus a constant.
so this one is morally right but wrong due to a typo.
probably the 41% of you, a lot of you didnt realize the plus c but you said ahh its none of that crap.
so you took that answer anyway so its sort of a free gift.
but thats okay.
so its important to remember not only as a clue to you
10:00what you are integrating but also as you'll see in a minute but something else that helps you keep track of whats going on.
so this role so in math we dont write something if its meaningless.
when we write notation here every little symbol in here tells us something.
the numbers here tells us what our bounds of integration are.
the big stretched out s tells us we're doing an integral.
this tells us our function.
10:30this tells us our variable.
you put a d in front of it so that we know its the differential of that its a rate of change.
a dx is not an x.
but theyre intimately related.
so all of these symbols here play a role.
so one of the problems that a lot of students have where a lot of people have in math is this is not true in writing prose.
in english, in..really in any language.
11:00even a very efficient compact writing language like chinese theres redundancy there.
theres no or very little redundancy here.
every little mark here on the page means something we dont just draw it to make it look pretty.
every piece of that symbol means something.
and so when youre reading math what a lot of students do when reading math they read all the words and then they say theres some symbol about something lets skip it.
you cant do that.
in fact what you should do is
11:31theres a bunch of words let me skip them let me look at the symbols.
thats really where all the content is.
there are words there to help you understand the symbols.
but both parts are important.
and theres a lot of redundancy in the words.
so probably you could understand this whole lecture so if you go home and watch the video and turn the sound off you could probably still follow this lecture just fine.
if you turn the video off and just listen to the sound youre going to be confused.
12:04by the way just im sure you all know that i set up those sneaky cameras in the back and i record every time these things i put on the class webpage so you can look at them later.
i dont know if you find it useful but i do it anyway.
so now i lost track of what im doing.
okay so we have all of these things meaning something now one of the first goals of the class
12:33is to look closely at integrals.
you know what integrals are from your previous class, well most of you sort of.
and now we're going to learn how to do a bunch of integrals.
so this is the last lecture in our review.
one of the main techniques of doing integrals so we already see that if we have so we know from before that-oh i wrote it there.
13:00that if we know the antiderivative then we know the integral.
whether its a symbolic one or a definite one.
so if we have something like the integral of.. the square root of xdx actually let me just so how many of you know how to do this?
okay so im not going to ask it.
so to do this we think of this as x^1/2.
13:35and now we just remember the power rule for derivatives and we turn it backwards so we increase the power by 1.
so 1/2+1 is 3/2.
and then we have to divide by this new power to adjust for the fact that when we take the derivative we want this to cancel that.
and then we add a constant.
so thats very easy.
if we have something slightly different
14:11so this is really morally the same question.
but its written in a different way that makes it look harder.
we use kind of a trick to change our point of view to make this easier.
and the trick that we use is we make a substitution.
14:34the substitution so that is what im talking about today in substitution what we do we write this not in terms of this x but in terms of a new variable that makes our life more convenient.
can someone tell me what that new variable is?
its u but tell me what i want u to be you can call it w you could call it joe.
so im gonna use w because you all said u i dont care what letter it is.
15:02we let w be 3x+4 because our life would be easy if this looked like that.
so we're going to let w be 3x+4 and then we just have now the integral of the square root of w dw but i did it wrong well in fact let me leave the dw off...this is wrong.
why is this wrong?
im supposed to have a 1/3.
15:30so im supposed to have a 1/3 because when i change x a little bit so if i change x from 1 to 1.001 w is going to change by 3 times that increasing x by .001 w is increasing by .003.
so thats the statement that 3 times the differential of w 1/3.
16:00dw is 3 times however much x changes.
take the derivative of these things only everybody did differential notation so that says when i wiggle x a little bit w will wiggle by 1/3 of that amount.
so this is the same statement you can solve this if you like.
and so here this is wrong
16:30because i left off the dx the dx here is a real part of the integral it actually represents something.
this part transforms to that but this part will transform to this.
the dx becomes dw/3 and the 3x+4 becomes the w.
and now this is easy this is just the same integral i did before.
so this is just 1/3(2/3w^3/2)
17:07the constant can go inside or outside because its an arbitrary number if you multiply an arbitrary number by a third you get another number.
so thats why.
so we can write this as 2/9... and we can go back here to turn the x because i may not have done that now again this should be review for everyone is anyone confused by this at all?
17:38what i want to emphasize is that we need to pay attention not only to how the stuff inside changes but how the differentials change.
i guess a little more formally whats really going on here
18:01is we're just using-we're just writing the chain rule in integral form so if we know that from calculus from first semester so we know that the chain rule says.. that if i have one function inside another function and i take the derivative
18:33then this is the derivative of the outside function plugging in the inside times the derivative of the inside function.
if we want to write this in differential notation this would also say... well this is good enough for what we need.
So thats the chain rule.
and now lets just integrate both sides of this thing.
if i integrate
19:06with respect to x then its still true and here my g is well in this example is w you can use u, you can use joe, i dont care.
so this is saying i have a function with something written inside it.
19:31so f in this example is the square root.
and g is the x+4.
and here this g'(x)dx this is a 3dx.
and which is the same thing as so that means if i have to write it in terms of w id write it as 1/3dw.
yes? no?
okay so this is just rewriting the substitution rule in a slightly more formal thing lets uhh do another example so suppose that i have the integral of
20:34of.. e^5dx so this should be easy.
what is this?
this is 1/5e^5x + some constant.
and here what im doing behind the scenes -i dont, i just
21:02i didnt write it down and you dont see that immediately but what im doing behind the scenes is im letting u=5x.
so du is 5dx.
which means that dx is 1/5 its du/5 and so when i changed this to that then i have the integral of 1/5e^u du
21:31and i just write it down because thats what i know.
so if you cant just look at this and write this down please put this intermediate stuff down until you get good at it.
i think most of you are maybe to that point where you dont have to write it down but maybe not.
okay suppose i have something like uhh and integral of.
22:01sin(x)/1+cos^2(x)dx what would be a good substitution to make?
okay so lets make this be a clicker question.
i should substitute
22:37u=sin(x) A B) u=cosx c) u=1+cos^2x d) for another class i should substitute for another class
23:07so write substitution as one of these things.
okay so im going to stop this even though there are people still madly punching their buttons.
okay stopping now.
stop answering!
okay here we go stop stop.
23:31okay soo a small amount of people, about 9% think this, theyre wrong.
well you could-so this would be hard.
you could do this but it would be hard.
in fact you do sort of all of them except well nobody chose this one okay fine.
so it really comes down to these 2.
if i did-so lets see what happens.
24:05when i try so here i have the integral sin(x)dx/1+cos^2(x) and if i make the substitution u=the cos uhh.. lets do the 1+cos^2x.
so the most popular choice but its close its 41 49%
24:32so those are close.
so if i make the substitution u=1+cos^2x then du is going to be the derivative of this.
which will be well thatll divide gives you a 2cosx times the sin of x dx..negative.
and so
25:00you know this is sort of the most complicated bit here.
so this becomes now i have a sinx dx and a u here then i have a sinx dx then i have this sort of i dont know quite what to do with this cosx business.
i have sort of a cosx left over that i dont know what to do with.
now i can work hard and try to figure it out
25:30but this is-so so i sort of have -2cos(x)du and i cant get rid of this.
so maybe i can make this go somewhere with a lot of sweat but i dont want to work that hard.
so lets try the other choice.
26:00so im going to let w be cosine.
and when w is cosine dw is -sin(x)dx so thats good because i have sitting around here a sin(x)dx.
so this is really dw except i just have to make it negative.
when i do this i see dw sitting there looking at my face.
so that means that this becomes
26:34then our top becomes dw when its negative .
and the thing on the bottom becomes 1+w^2.
now this is an integral that probably you know how to do.
should know how to do.
if you were sleeping when they did inverse trig functions, well then you forgot.
but, this is the arc tan.
so this is the arc tan of w.
27:06then our w was cosine cosine x plus a constant.
so really answer B is the better answer but its not obvious.
27:32so one of the things about substitution is its not completely obvious what the right substitution is.
its more obvious when you play with it a lot and get some background but its not completely obvious.
so it requires a little effort and maybe sometimes a little bit of a wrong turn.
but its okay to make wrong turns as long as you eventually get it right yeah?
28:04yeah so youre saying its 1+3w^2 here?
and this is what i want to integrate?
so here i would make the substitution the u is.. well see i want this to look like 1+ u^2.
28:33so i wouldnt make the substitution u=3w.
i want u to be the square root of 3w.
because now this looks like 1+u^2.
so then du is.. root 3 dw so i pick up the fact that 1/root3 dw
29:07so this is 1/root3 arctanx k?
a root 3 though.
want me to finish writing it?
so its a 1/3 square root yeah?
29:30yeah im not saying this wont work.
its too hard.
so really B is not wrong.
sorry C is not wrong.
i have trouble with letters.
c is not wrong.
its just hard.
so i just gave up and i just went ahhhh!
because this came out you could maybe make this work.
but if i ask you
30:00to drive to port jefferson or walk to port jefferson its not wrong to go to huntington first.
its just not really a good idea.
you could maybe make this work.
so in some sense and maybe you could even make u=sinx work although i dont see how.
so in some sense theres no completely correct answer to this its just that the best substitution is certainly
30:33the cosine.
so substitution i want to emphasize here that by analogy i said this before i guess ill use this board again.
31:02differential form dx, whatever we're calling it when we write this thing which everyone always seems to forget, not everyone but many people seem to forget plays 2 roles.
this dx it tells us the variable
31:34but its also like a unit.
its telling us when we make a substitution its telling us how to transform the substitution.
so it acts like a unit or units of measure
32:00by analogy this is like saying this one this one is like the dx is saying inches.
or meters or pounds its a unit.
and when we transform it into some other unit we write this function in terms of some other variable u what have to say what the relationship is between the original quantity and new quantity.
and thats all the chain rule is saying too.
32:31the chain rule is saying if we measure-if our g is in feet and we change it to inches i have to multiply by 12.
now sometimes in math not so much in science the difference in the units are not linear.
sometimes in science so for example uhh when youre measuring an earthquake the energy is related to the value on the richter scale logarithmically
33:01so the change in units there is non-linear but typically in science and life in general most of the units are related by a linear relationship.
not so here in fact the linear ones we can do in our head the non-linear ones we write down - someone had a question earlier.
you gave up.
didnt you raise your hand?
somebody over here was raising there hand and i was talking so i ignored them.
33:34okay sorry i ignored you.
and you forgot your question.
right so this one is a non-linear change in variables.
did i have another one i was supposed to do?
i mean and some say all of the substitution problems are the same.
34:00once you get the hang of it, youre done.
so and supposedly you know that already.
so i think im not actually well okay so the idea in substitution so you wont know of course whether its a substitution problem or not.
but when youre doing a substitution
34:31you want to look for uhh let me call it u so you want to let u be something and the thing that you want to let it be is something that number 1 simplifies the integral.
but also number 2
35:01the derivative is laying around in some sense.
and what i mean by that so for example in this guy letting u be 1+cos simplifies this integral but the derivative is not here nor is it easily made to me.
letting u be cosine
35:31simplifies the integral to be a harder integral but its derivative is right here.
let me do one more example.
36:03and the homework thats due next week well you'll see theres not always a unique substitution.
alright if i have the integral lets do uhh..
36:31suppose i have an integral like that.
and im purposely not using an x.
i cant use u for my substitution.
so i shouldnt use u.
i'll use x or i can use whatever, i'll use x.
so whats the obvious substitution to make? yeah?
if i let x be u^2+1
37:00then dx is 2udu and i have a u du laying around.
so thats good so then this just becomes square root of x dx...1/2.
theres the 1/2 dx and so this we already did this.
37:30this is 1/2 of.. this is x^1/2 so this becomes 3/2 so this is 2's cancel, 1/3(u^2+1) to the 3/2 plus a constant.
38:12okay so this example was a stupid one cause i remembered the wrong example.
this happens.
what was the one i wanted to do? i dont know.
oh that one. sorry.
38:32okay so that was easy.
this was an easy one.
i should have done a harder one but thats okay. so a lot of integrals, in fact most integrals are attackable
39:00by this technique of substitution.
so what if i have something oh i guess i need to say something about the bounds sorry.
so i forgot that.
suppose i have a definite integral.
uhh for example an integral from 1 to 5 of... (ln2x/x)dx.
39:33suppose i have some integral like this that i need to do.
for whatever reason.
then i can do this in 2 completely equivalent ways.
and i prefer the first one but many people would do the second one so let me do it both ways i suppose.
theres an obvious substitution here.
someone want to tell me what it is?
so if i let u be the natural log of 2x du except for a factor of 2 is sitting around here.
because i remember that the derivative of the log is 1/x.
so here if i let my substitution be the ln2x then the differential is 1/2x
40:30times the derivative of 2x which would be 2 times dx.
and so this is just 1/x dx which is sitting right here.
so this just becomes the integral of u du and i did this last time and some people objected this says x is 1 and this says x is 5 we just dont usually write it.
41:01these numbers if i write 1 to 5 here this is wrong.
because this is saying u is 1 and u is 5.
its wrong.
so well actually we'll do it the way i dont like first.
the way i dont like i can write for myself x is 1 and x is 5.
then i dont forget.
that ones not wrong its just unconventional.
so because i have to remember what x is.
41:31but okay.
so if i do this then this is 1/2 of u^2 between x=1 and x=5.
which gives me well i have to remember what x is, x is the log of 2x.
no u is the log of 2x thank you.
so this is
42:001/2(ln2x)^2 from 1 to 5 i dont have to write the x anymore because its the same.
and so this is 1/2(ln10)-1/2(ln2).
what you can simplify a little bit is the log of 10/2 which is the log of 5.
42:30if you want.
which is the same as the log of the square root of 5 if you want i dont care.
theyre all the same thing.
so thats one way you could do the problem.
which has a lot of extra garbage in it.
so what i think is a slightly more efficient and also a mentally easier way to do the problem
43:00is to just forget about x once you change to u.
its really the same so here i have the integral from 1 to 5 ln2x/x dx i make the same substitution.
and notice that when x
43:31is 1 u is the ln2 and when x=5 u is the ln10.
i dont have to write this down i can just think it.
and so this becomes then the integral from the ln of 2 to the ln of 10 of u du 1/2u^2 evaluated from the ln2 to the ln10.
44:01which is just 1/2..wait did i lose a square somewhere?
i sure did, why did nobody stop me?
so this is all crap we'll fix it in a minute.
uhh (ln10)^2-(ln2)^2 there.
this ones right now.
and this one needs to be adjusted.
44:30and this is garbage.
let me get rid of the parentheses.
so now theyre both right and theyre both the same.
this way is a lot more efficient in my mind and its sort of what you do anyway if youre working with units.
suppose youre doing something in feet and you discover that its easier to work in inches.
you dont change to inches
45:01for part of your calculation and then change back to feet and blah blah.
you just leave everything in inches, the numbers the same, were done.
so i just changed to a more convenient measurement system and then i just changed everything.
so when youre doing definite integrals you can treat them as indefinite integrals do it all out, and then plug in at the end or you can just change over.
just convert youre line to the new system you get the same answer.
45:32or you get an equivalent answer because youre doing the same thing.
what about... so ive been doing substitution integrals all day long but im tired of them and you probably are too.
46:03suppose i have an integral that looks like this.
is there a good substitution i can make?
so some of you know, and youre saying no.
some of you dont know and youre just looking blank.
theres no good substitution i can make here.
46:30i cant find a good substitution so i have to be a little more clever.
i want to transform this into something else.
now when we came up with the substitution rule we actually looked at the chain rule.
chain rule is an important part of this branch of calculus.
theres another rule that you learn when youre learning derivatives that is extremely important.
the product rule.
we didnt use the product rule but maybe the product rule will be our friend here.
47:03i can integrate that piece i can integrate this piece.
i would like to write this as-so this is not true.
this is wrong.
and if you try that, you fail, youre done, go away.
this is dumb dont try that because its just wrong.
but we can manipulate some kind of product rule.
47:34so lets remember the product rule which is for derivatives.
and we want to somehow get a product rule for integrals the product rule for derivatives says that if i have one thing multiplied by another thing and i want to take its derivative
48:01well then i can write this as the derivative-i know im mixing notation but thats okay.
the derivative of the first thing plus the first thing times the derivative of the second thing.
soo maybe i can integrate this and get something useful.
so if i integrate
48:31everything in sight this is certainly still true
49:05so now remember im trying to do the integral of xe^x dx.
and as you know if i dont think of this as 2 things that i want to integrate separately i could maybe think of one of these as being the derivative and the other one as being an integral that is when i integrate e^x i get that.
49:31so i could have this-think of this as the derivative of something.
when i integrate x it is more complicated when i take the derivative of x it becomes a 1.
so i could restructure this guy to say that well i want this if i have something like this
50:09since i have an equal sign i can subtract this thing here.
and this is e this is the integral of the product and this is the product.
this is the integral of the derivative.
50:33now theres a more familiar form that many of you may have seen this before.
that is if we think of and unfortunately theres an unfortunate notational collision.
if we call one of the functions u and the other one v this says that the integral.. lets let f be u instead.
the integral of a thing times the derivative of another thing
51:00is the same as the product of those two things minus the integral so i can transform i can trade a product of something i can take the derivative of and something i integrate into the things and..
51:30integrate that guy and take the derivative.
now this is more youve never seen this before so this is called integration by parts.
maybe youve seen before maybe you havent but we will spend a little more time on it.
let me just do this xe^x example then we'll call it a day.
so in this case
52:01if i let if i let u be x then du is just dx.
and if i call this part my dv because i can integrate it then when i integrate v is just e^x.
so this is not a substitution and its unfortunate that u
52:31is used typically for both substitution and for the parts of- integration by parts but thats the way it is.
and so that this formula over there tells me that this integral is now the product of this and this.
minus the different integral product of.. v du this and this.
well this integral is easy.
53:13so that made something that was hard into something a little easier it takes a lot of practice to get used to picking out whats what just like it takes a lot of practice to figure out whats what in u substitution.
this is not on the homework thats due wednesday.
53:30so if you dont understand it yet dont sweat it out.
okay have a nice weekend see you monday.