Just to clarify: If you’re in Calc 1 or 2, please write down your steps. This video was geared towards Calc 3 students. Sorry if I came off as condescending, it wasn’t my intention at all
As I just finished calc 3, I agree it's a waste of time to write all that down, I think a reasonable amount to write down is u= and maybe du= that's it. Any more and it's just wasting time. With the two things written if you mess up you will still get partial credit and you have time to still do the second integral or whatever else.
Counterpoint: you can’t get partial credit for work done in your head! In any given calculus class in 2021 American schools, a solid 75% of the class will get almost all the answers on a given test wrong and they only survive the class at all by showing their work and getting partial credit
@@nedmerrill5705 True, but it works the same way. One wrong move and you lose in chess. However, if there were a long series of moves made before that one that were great, then this demonstrates more about the player than if there were not such a series of moves played. And this is what credit should really be based on - the potential the person has to make great moves in future situations. In math, showing a lot of good work but getting the wrong answer is more predictive of you getting right answers in the future than having no correct work does.
I wouldn't phooey the students who write down all the steps during u-sub at all; it's a great way to catch mistakes, which is especially important in an exam setting.
Yea I get really bad anxiety in exams because of the time limit. I've been known to divide 80 by 4 and get 10 or something stupid like that during big exams, so why would I not write everything down so I can easily check it at the end?
forget the time limit; the u-sub is not intuitive at first. they need to get used to the substitution before they do it secondhand. it's the same with definition of derivative. you use it only once to do a proof and then you just apply the differentiation rule.
I feel like I would make more mistakes doing u sub than doing it the second way. It's easy to confirm that the derivative of the function he wrote is correct.
Amen. Last test i was trying to simplify some things for a calc exam and i took 1/2 on one side and transferred it to 1/2 on the other instead of multiplying by two. I am in Calc two. Just an idiot
Also profs not giving full credit because you didn’t write all the steps! My professor cut marks because I didn’t write Boolean algebra is distributive :)
since simple u-subs were taught at the end of calc 1, and were considered review in calc 2, when i got to calc 2 my teacher said 'any time you are u-subbing a linear function you dont have to write it as long as you remember to divide by the derivative', and he called it a "lazy sub". by the time i got to differential equations, the professor didnt really care about showing work for integrals, so i would also do a "lazy sub" whenever the integrand is of the form [ f(x) ]ⁿ * f ' (x)dx. probably the most common one is ∫ f ' (x)dx / f(x) , = ln | f(x) | + C, which is sometimes just taught as a formula for anything more complicated, i typically dont trust myself to not make mistakes if i dont write out the u-sub steps. they are pretty quick anyway
@@DoctrinaMathVideos i always hate when teachers say to "show your work" but dont specify what that means. what exactly needs to be shown, and what can be done in your head? does moving terms to one side of an equation and then factoring something out count as one big "algebra step", or do i have to write each line separate? what if in my head i automatically do something in one step without thinking about it because its second nature? do i have to draw a number ling and count markings everytime i want to add two integers? theres just too much ambiguity. i absolutely agree that showing your work is important, but all im saying is that you have to be very precise about what that means or it will lead to confusion.
@@nathanisbored lmfao so true - I even ended up showing properties of various math stuff like properties of logarithms, properties of trigonometric functions, properties of exponents and fractional exponents, deriving the formulas of how those equations are formed and also showing mathematical axioms of basic arithmetic :'D it's so obssessive and compulsive XD
I agree, essentially apply the amount of "laziness" you are comfortable with and fall back to the algorithm when you need more confidence. Nothing wrong with applying a robust algorithm, I don't think that makes you any "worse" at math.
Wow in france we never do substitions where not needed. You know for example f'*f is the derivative of f^2/2. By definition you can rightaway tell it's your primitive (up to a constant). So we have to train a bit to recognize them but it's often straightforward (sometimes less obvious like 1/cos, but you just have to write it as cos/1-sin^2) Im glad we use substitutions only when it is helpful (for example Bioche's rules)
In the UK, this method is called the "reverse chain rule" (I'm not a big fan of the name though). It's one of the first integration techniques students usually meet - I didn't realise it's not such a big thing everywhere else!
@@SatyaVenugopal exactly, but integration by substitution is just a formal method for reversing the chain rule whereas this way is a more intuitive way of doing it
I haven't seen anyone mentioning "puting everything under a differential" instead of a u-sub, so I guess I'll mention it. If you have an expression f'(x)dx you can substitute it with df(x). Example: ∫cos(x)²sin(x)dx=-∫cos(x)²dcos(x)= -⅓∫dcos(x)³=-⅓cos(x)³ That's it. You don't have to write it down every time, you can do it in your head. And at no point any guessing is involved. I think this small trick (not really a trick, it's completely rigorous) could be of some help to someone.
@@chessematics yup I've got some recent calculus books from American authors and the name u-sub is very irritating I much prefer books like Spivak's calculus,Tom Apostol or Russian books like Piskunov,Berman and Demidovitch
I definitely understand your point dr. Peyam, but I also understand that these insights and "tricks" are acquired over time, not just in a semester calculus course (especially when students come from ''poor'' mathematical foundations and have little knowledge). Over time they will realize this. It may be good practice to encourage them early on, but I think it's unforgiving to require this of them, especially in exams and assessments.
…why are you guys talking about it like its a big deal It’s literally just an integral of the form f(ax+b) It is well known that this can be substituted and its well known this can be directly integrated because of the fact the substitution’s derivative yields a constant anyway.
If it’s possible, the area function A(f(x)) can be thought of as F(x) scaled by a factor of X(u) assuming there’s some manipulation matrix M that can transform f(x) into g(u(x))/u’(x). In that case, you could define all definite integrals as u substitutions with constant u functions. Meaning, all definite integrals are really indefinite integrals with very boring input functions.
What you are actually doing *is* substituting mentally without writing it down, I do that all the time. But I know my students are not yet conmfortable with it, and I ask them to write down their reasoning -:either for them to get partial points when the end result is false, or for me to check that good answers are not the product of an error (or copying from a neighbour).
Nothing wrong with using u-sub. Keeps one from making a mistake under the pressure of an exam. I wouldn't have done it but others who don't think in their head very well or are afraid not to show their work may need to.
Agreed! I did u-sub the first 1-2 days of when learning integrals but then it just clicked for me and I barely write it down. If someone needs to do it, please continue to do so, but don't be afraid to try it without as well!
Totally agree ... the u-substitution method is logically sound and promotes error-free work/steps in performing anti-differentiation ... plus, it reinforces an understanding of the four fundamental parts of a (single) integral: integrand, differential, upper limit, lower limit
Students who are starting Calculus I and beginning to know what integrals are, PLEASE just do the long process, you will better understand your problems and if you feel you grasp perfectly the patterns, then look at this video. It’s like playing chess, you don’t begin learning by applying 4 moves of tactics when you first need to understand well how the pieces moves, same with math.
@@drpeyam perhaps if you like, you can add like a disclaimer in the description? Then the students who understand the idea can develop and get comfortable with this quicker process which saves them space and time.
We learn this in England: its called integration by inspection or informally, the reverse chain rule where you consider the antiderivative, differentiate it and then adjust for the constant. However with enough practice you can do it in your head like you are 👍 Edit: We also learn u sub as a separate method.
In fact, my textbook introduced this sort of Reverse Chain Rule before introducing substitution as: if ∫f(x)dx = F(x)+C then ∫f(ax+b)dx = 1/a F(ax+b) + C With the other examples like 1-x^2, we had to do the substitution. Great video btw Peyam!
I would argue this is useful in exams only. As you said in the end the problem is not using the u-sub, the problem is writing it down every time. When still learning it is not obvious what the antiderivative is. At 3:40 for example, someone learning would have to guess between x and (1-x^2). Doing the U-sub step by step trains you in recognizing the anti-derivatives. You can do it because you have the repertoire, no one has it in the beggining of learning Calculus. All those "notice" are skills you already developped, they are the clever maths we develop while learning calculus. So I disagree with the title. The video is still nice, don't worry.
I mention to my students that the u-substitution rule is just doing a change of basis (a topic from linear algebra) or the chain rule for integration. I think textbooks that use the term u-substitution should just drop the term and call it the chain rule for integration or more technically the change of basis rule. In any case, it is a very important rule because it is incorporated into the other techniques in math especially in calculus 2 and differential equations. Edit: I have also seen it referred in some textbooks as the composition rule for integration just as the composition rule (chain rule for derivatives).
It's a little different from the change of basis (which is why some analysis texts call it change of variables), because you're considering differentiable functions rather than linear ones (different morphisms). Great point though!
I'll never forget doing integrals with my peers and being like "so what I do here is just kinda a backwards chain rule in my head" and someone said "... You mean u-sub?"
I am gifted in mathematics and still wrote down my u-subs throughout my entire Bachelor's degree. I like seeing all of my work in front of me. Not only that, but my developmental disabilities make it difficult to process information quickly and efficiently. Students should do whatever they are comfortable with and it's ridiculous to criticize the education system. Those that will benefit from not writing down will start doing it on their own.
"Students should do whatever they are comfortable with" "it's ridiculous to criticize the education system" How are these statements compatible when the education system doesn't let students do what they're comfortable with? The calc teachers I've met would take off points for doing integrals this way since the student isn't "showing their work".
That's pretty much how we learned integration in the UK. You'd use substitutions for more complex stuff (generally the sub would be given) but you would do normal problems just like this.
U substitution is more standard especially for the beginners. Because it gives them a clear look to what the entire steps look like. But having known the entire process already, it's recommended to avoid Substitution in this case.
sin(x)dx looks like d(-cos(x)), and we can move "minus" sign outside differential and integral. So we get -Integral(cos^2(x) d(cos(x))), which looks like -Integral (something^2 d(something)). Which is obviously -something^3 / 3 + C, i.e. -cos^3(x)/3 + C
I was taught this a couple weeks ago in my calc bc class, it's called "license to integrate" where you have the derivative of the inside on the outside and you can just ignore it and differentiate the inside but a much cooler name that my teacher came up with was based of the album "license to ill"
This is literally just how I was first taught to solve integrals at school, to see if you can recognise the antiderivative. It always frustrated me how I never saw anyone using it since it’s so much simpler and more intuitive.
Well it takes much more practice in normal derivatives and most students jsut dont have it so it’s more error proun for them hence why they just use a u sub which is actually cleaner at least in my opinion
Is there a particular reason why it is called the Lambert W function? Lambert is obviously the name of someone involved in coining the term, but what does it have to do with W?
That's literally what I've always been doing. I've always been in pain seeing my friends go through the whole process of u-sub, while I just do this (which I call "reverse chain rule") in my head. Thanks for making this video!
My calculus teacher in high school taught us this method. It was especially useful in evaluating integrals involving the power reduction for trig squared functions
I usually like to do a shorthand u-sub: For example, because d(-cos(x))/dx = sin(x), cos²(x)sin(x)dx = cos²(x)d(-cos(x)) = d(-cos³(x)/3) Also eliminates the fuss of having to calculate u(t_0) and u(t_1).
I was always really bad at figuring out when we were given problems in which we were *supposed* to use u-sub (I simply didn't recognize if it was supposed to be needed or not) so I just usually did the problems like this, since it's a natural continuation of differentiation
i feel like these kind of method is useful only when you are exposed to differential equations. sadly in our country the order of math sub for calculus as a whole are as follows: 1. differential calculus 2. integral calculus 3. differential equations my point is.. doing U-sub is a standard thing to do.
I would say that this depends on your teacher. I've gotten marked down over 14% on one problem for doing the bulk of it in my head. They want to "see the process."
Interestingly, I stumbled upon this technique back in college ages ago when I ignored my professor (who simply said no) and asked the question: are there any functions other than y = e^x whose differentiation is equal to themselves? I thought about it for a long time and constructed y = (ln(x))^(x/ln(x)). This is of course pretty close to the definition of e^x but with a restricted domain, but I ended up calling the method integration by construction and was happy with the results as I learned to apply the process to other integrals. Great video!
What you're referring to as a "restricted domain" simply means a domain for which the algebraic expression is not well defined on part of the domain. But on the portion of the domain for which it is well defined it is identical to e^x (Identity of indiscernibles). Your example is not particularly special in that one can always create a function identical to e^x but not well defined on a set (to do this for a point a, consider e^x * (x-a)/(x-a)). We normally exclude functions that are not well-defined on the expected domain precisely for that reason.
Thanks for the tip! I've always thought that u-subs could be avoided, but when it came to actual problems I wasn't clear on how to go about it. Seeing it done makes the method more obvious.
So essentially this trick is intuit the antiderivative instead of applying the change of variables formula? And the argument of the video is you should be good enough at this stage to intuit what the antiderivative is
As many have already stated, I have seen professors take as much as 20% off of all correct answers simply for not writing that the words of the convergent/divergent test they used even though they showed all the work required to utilize said test for the problem. I also think it's important to always show why a particular technique used works. Writing down the steps to use during u-sub helps reinforce in my head why it works. I know that every professor has their own particular preferences about how they want to teach, but this is the first time I've ever seen a professor show some displeasure with too much work shown? And as a tutor for calculus, I don't know how hard it has been to help a student dissect their incorrect answer when they've simply done steps in their head just to save time.
Whenever I teach Calc, I make sure my students add the following rule to the list of integration rules given in the textbook, which I call the Linear Anti-Chain Rule (or, more verbosely, the chain rule for antiderivatives with linear inside functions): Int(f’(mx+b)dx) = (1/m)f(mx+b) + C
What about finding an antiderivative of (e^x+xe^x)(1+ln(xe^x))? In this case, it seems natural to simplify the problem by first taking u=xe^x. Also, the substitution method might be used later in formal derivatiions of solutions to PDE (especially when the coefficients are allowed to be distributions). Perhaps you might also use the substitution rule to cleverly rewrite a PDE prior to application of numerical methods in order to get faster convergence. This is similar to the fact that the integration-by-parts rule has much more significance than it first appears. Without using it, I don't know how one would define the definition of a distributional derivative nor prove that it's equivalent to the standard derivative for distributions arising from functions.
*Dr. Peyam,* in the last example you could just factor out (-6)^3 out of the integrand so that you would be dealing with just (x - 2)^3 and the new constant multiplier in front of the integral. Then the antiderivative would follow smoothly.
Only really works when integrating gives something of the form k*[f(x)]^c though. U substitution is the best method for lots of integrals such as sqrt(c-x²)
I didn’t learn integrals in my calc class yet but despite that this was so easy to understand with only knowledge of derivatives. Double integrals look pretty scary lol
As a half way point you can write it as an integrand and measure often so integral A * f * df/dx dx we see this as A/2 f^2 immediately better we can often see an adjusted chain rule directly in the integrand and again seperating the parts into the constant the chained integrand and the relative measure makes this clearer.
when it comes to writing steps out in calculus especially, I found that everyone can do it in a different number of steps. The amount you do is determined on how well you do math in your head, trust your teacher to get it, and how much you trust yourself, which is why I write it out all the way until my hand cramps.
I think the reason why substitution is a thing is because it is an example of the pull back of differential forms under the transformation from x to U. So in case if you somehow want to study math in the future. you have already touches some examples of the pull back map
I go to a UK uni and my US friends would sometimes ask how I did an integral in one step instead of 2 u subs or something. Using the reverse chain rule is so much faster
u basically asks to think chen lu in reverse. analogically to diffrenciate a composition w/o u-sub e.g, differentiate [(2x+1)^2], don't write f(u)=u^2, u=2x+1=>f'(u)=u^2,u'=2...
Surprised no one is mentioning you can just expand the polynomial and then easily integrate each term. No u-sub, no dividing by derivatives of linear factors. Obviously this gets trickier the higher the value of the exponent, but for linear terms that are squared or cubed, just expanding the polynomial out is very easy.
Maybe that's something that is mainly taught in English-speaking countries. In German high schools (Oberstufe) we were taught these integrals without assigning the "rule" any name.
Hi! This is the method which we learned in high school, but now in uni where we start with "proper" u-sub I've run into some troubles. I'd like to continue doing this as it feels more intuitive, but I've noticed that some integrals are just impossible for me to solve "in my head" since I cannot only multiply/divide by a constant in order to achieve the inner derivative in my integrand. For example: e^sqrt(x) I understand that the inner derivative is 1/2 * x^(-1/2), but since I don't have any factor of x in my original integrand I don't understand how this is possible to solve without the "proper" u-sub. Moreover, I don't understand why it works with proper u-sub if u-sub truly just is the reverse chain rule. Any help would be appreciated!
I thought in an exam you to show your steps and the answers may only count as 1 or 2 Marks, whilst the methodology counts for most. If I was marking you I would deduct marks for taking shortcuts.
Shouldn't this 'U substitution' just be called reverse chain rule or chain rule for integrals, given that it is usually used for 'nested' functions like log( (3x + 4)^8 ) ?
Dear Prof. Payam, the problems are two: 1 - it is not teached to memorize generilized immediate integrals; 2 - no model is taught, because of using memory requests too effort. The conclusion is that student think that they can study math without memorizing, which is impossible. Greetings from Italy.
You can lead a student to knowledge, but you can't make them think. Any student can memorize a method, but some just don't have the intelligence to use these kinds of shortcuts.
What surprises me is that even among the good students who use the u-sub correctly in the more sophisticated integrals (more like Calc 2 stuff), don't really know WHY the u-sub actually works. And when digging into the textbooks, apart from a formal proof (typically in some appendix) there is no proper (geometrical) explanation as to why it actually works. Sadly...because in essence the u-sub is a "reverse" operation from the chain rule. At least, that is how it could be introduced in Calc 1. But most students learn the u-sub as a "step" problem step 1,2,3 and done.
If you are needing to demonstrate to someone else that you know what you are doing it makes sense for them to ask you to show all steps. When doing it for yourself of course you don't need to.
No. I am good at calculus. I'm studying for my multivariable calculus final exam right now actually. BUT when the problems are more difficult than the simple stuff here and the number of steps and processes to get your final answer increase, it takes more time to write out your steps but saves in the time you would spend double tracking to check why your answer is wrong. And for an exam, you don't have that luxury. If you're wrong you're wrong, but now you don't get partial credit. I also tutor in calculus by the way and I NEVER recommend skipping steps like that. I sometimes do, but only when I don't have any doubt about the result. People still learning should not be doing things like that, because it will just make it harder.
So the reason I’m mentioning this is because some of my multivariable students spend 5 mins writing out a simple u sub, which makes them lose valuable time, especially if they have to evaluate 4 double/triple integrals, that’s why I recommend them to do it mentally
@@drpeyam I get that, but from the students perspective it is way too hard to check my work when I get done doing all those integrals. It's too much to keep in my head, even tho I've had a ton of practice. Sometimes I can do it but most of the time I can't trust myself with something as high stakes as an exam. One mistake is all it takes. For me the most important thing is book-keeping.
After learning about the reverse chain rule in my calc class after self learning U sub from calc channels, I noticed that U sub is a cumbersome crutch. However, the related skills of U sub do carry over towards trig substitutions, but that is the only one I could think of that could be useful.
I like trying to do the steps implicitly in my head, but sometimes the Algebra ends up tripping me up. Like I'll forget a constant factor of pi or something because it pops out of the Algebra about the du, or I'll miss a sign or mis-multiply when evaluating the integral. Since we are not electronic computers, we shouldn't make becoming one a goal.
:-) Dear colleague Peyam - doing multiple steps at the same time in your mind - it might create mistakes (divide or multiply by 2/3 ??). So, if I want to be sure that I'm correct, even I prefer to insert an extra step (may be I'm a bit prone to stupid mistakes)
Did you misspeak when you said "multivariable exam?" That question is clearly from Calc 1 or Calc 2. Also, the method you used is basically how kids are taught that they need U-sub. PLUS U-sub, the method not only is usful for this, but also get's students used to using the letters U (sometimes V too) so when they do IBP, they'll be used to seeing UV-int(vdu).
It doesn't sound very rigorous. It sounds like just pulling the answer out of nowhere by magic and looks like the questions are contrived to work well with the method.
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Just to clarify: If you’re in Calc 1 or 2, please write down your steps. This video was geared towards Calc 3 students. Sorry if I came off as condescending, it wasn’t my intention at all
lmao
It's good clickbait though, don't feel too bad...
We learnt this before learning u-sub.
As I just finished calc 3, I agree it's a waste of time to write all that down, I think a reasonable amount to write down is u= and maybe du= that's it. Any more and it's just wasting time. With the two things written if you mess up you will still get partial credit and you have time to still do the second integral or whatever else.
We refer to this as reverse chain rule
Counterpoint: you can’t get partial credit for work done in your head!
In any given calculus class in 2021 American schools, a solid 75% of the class will get almost all the answers on a given test wrong and they only survive the class at all by showing their work and getting partial credit
Right. But there's no partial credit in chess.
that's very telling, isn't it?
Ned Merrill So deep!!
@@nedmerrill5705 Mind. BLOWN.
@@nedmerrill5705
True, but it works the same way. One wrong move and you lose in chess. However, if there were a long series of moves made before that one that were great, then this demonstrates more about the player than if there were not such a series of moves played. And this is what credit should really be based on - the potential the person has to make great moves in future situations. In math, showing a lot of good work but getting the wrong answer is more predictive of you getting right answers in the future than having no correct work does.
I wouldn't phooey the students who write down all the steps during u-sub at all; it's a great way to catch mistakes, which is especially important in an exam setting.
Yea I get really bad anxiety in exams because of the time limit. I've been known to divide 80 by 4 and get 10 or something stupid like that during big exams, so why would I not write everything down so I can easily check it at the end?
forget the time limit; the u-sub is not intuitive at first. they need to get used to the substitution before they do it secondhand. it's the same with definition of derivative. you use it only once to do a proof and then you just apply the differentiation rule.
I feel like I would make more mistakes doing u sub than doing it the second way. It's easy to confirm that the derivative of the function he wrote is correct.
Amen. Last test i was trying to simplify some things for a calc exam and i took 1/2 on one side and transferred it to 1/2 on the other instead of multiplying by two. I am in Calc two. Just an idiot
Also profs not giving full credit because you didn’t write all the steps! My professor cut marks because I didn’t write Boolean algebra is distributive :)
Dear students, pls use anything you are comfortable with (even u-sub)
"How the u-sub skip DESTROYED Integral Any% Speedruns"
New summoning salt documentary coming up
Speedrunning and calculus? This is my favorite crossover. TAS when?
I actually have an integral speedrunning competition tomorrow, so this applies to me
since simple u-subs were taught at the end of calc 1, and were considered review in calc 2, when i got to calc 2 my teacher said 'any time you are u-subbing a linear function you dont have to write it as long as you remember to divide by the derivative', and he called it a "lazy sub".
by the time i got to differential equations, the professor didnt really care about showing work for integrals, so i would also do a "lazy sub" whenever the integrand is of the form [ f(x) ]ⁿ * f ' (x)dx.
probably the most common one is ∫ f ' (x)dx / f(x) , = ln | f(x) | + C, which is sometimes just taught as a formula
for anything more complicated, i typically dont trust myself to not make mistakes if i dont write out the u-sub steps. they are pretty quick anyway
I tell my students to always show their work. It provides more practice and it demonstrates mathematical maturity.
@@DoctrinaMathVideos i always hate when teachers say to "show your work" but dont specify what that means. what exactly needs to be shown, and what can be done in your head? does moving terms to one side of an equation and then factoring something out count as one big "algebra step", or do i have to write each line separate? what if in my head i automatically do something in one step without thinking about it because its second nature? do i have to draw a number ling and count markings everytime i want to add two integers? theres just too much ambiguity.
i absolutely agree that showing your work is important, but all im saying is that you have to be very precise about what that means or it will lead to confusion.
@@nathanisbored lmfao so true - I even ended up showing properties of various math stuff like properties of logarithms, properties of trigonometric functions, properties of exponents and fractional exponents, deriving the formulas of how those equations are formed and also showing mathematical axioms of basic arithmetic :'D
it's so obssessive and compulsive XD
I agree, essentially apply the amount of "laziness" you are comfortable with and fall back to the algorithm when you need more confidence. Nothing wrong with applying a robust algorithm, I don't think that makes you any "worse" at math.
Wow in france we never do substitions where not needed. You know for example f'*f is the derivative of f^2/2. By definition you can rightaway tell it's your primitive (up to a constant). So we have to train a bit to recognize them but it's often straightforward (sometimes less obvious like 1/cos, but you just have to write it as cos/1-sin^2)
Im glad we use substitutions only when it is helpful (for example Bioche's rules)
In the UK, this method is called the "reverse chain rule" (I'm not a big fan of the name though). It's one of the first integration techniques students usually meet - I didn't realise it's not such a big thing everywhere else!
Reverse Chen Lu!!!
@@drpeyam Much better!
I mean... isn't that what a u-sub is? The undoing of the chain rule / chen lu? (Just as integration by parts is undoing the product rule / prada lu)
@@SatyaVenugopal exactly, but integration by substitution is just a formal method for reversing the chain rule whereas this way is a more intuitive way of doing it
I'm from the U.S., and I also use the term "reverse chain rule" for what Dr. Peyam showed in this video.
I haven't seen anyone mentioning "puting everything under a differential" instead of a u-sub, so I guess I'll mention it. If you have an expression f'(x)dx you can substitute it with df(x). Example:
∫cos(x)²sin(x)dx=-∫cos(x)²dcos(x)=
-⅓∫dcos(x)³=-⅓cos(x)³
That's it. You don't have to write it down every time, you can do it in your head. And at no point any guessing is involved. I think this small trick (not really a trick, it's completely rigorous) could be of some help to someone.
I do exactly the same and my teacher yells at me 48 hours a day for doing this
That's how I was taught to do integrals and even the "u sub" was never u sub but the substitution method or change of variables
This is pretty cool!
@@marcioamaral7511 the name u-sub belongs to the American system, which is one of the most useless conventions ever.
@@chessematics yup
I've got some recent calculus books from American authors and the name u-sub is very irritating
I much prefer books like Spivak's calculus,Tom Apostol or Russian books like Piskunov,Berman and Demidovitch
I definitely understand your point dr. Peyam, but I also understand that these insights and "tricks" are acquired over time, not just in a semester calculus course (especially when students come from ''poor'' mathematical foundations and have little knowledge). Over time they will realize this. It may be good practice to encourage them early on, but I think it's unforgiving to require this of them, especially in exams and assessments.
…why are you guys talking about it like its a big deal
It’s literally just an integral of the form f(ax+b)
It is well known that this can be substituted and its well known this can be directly integrated because of the fact the substitution’s derivative yields a constant anyway.
If it’s possible, the area function A(f(x)) can be thought of as F(x) scaled by a factor of X(u) assuming there’s some manipulation matrix M that can transform f(x) into g(u(x))/u’(x). In that case, you could define all definite integrals as u substitutions with constant u functions. Meaning, all definite integrals are really indefinite integrals with very boring input functions.
What you are actually doing *is* substituting mentally without writing it down, I do that all the time. But I know my students are not yet conmfortable with it, and I ask them to write down their reasoning -:either for them to get partial points when the end result is false, or for me to check that good answers are not the product of an error (or copying from a neighbour).
Nothing wrong with using u-sub. Keeps one from making a mistake under the pressure of an exam. I wouldn't have done it but others who don't think in their head very well or are afraid not to show their work may need to.
Agreed! I did u-sub the first 1-2 days of when learning integrals but then it just clicked for me and I barely write it down. If someone needs to do it, please continue to do so, but don't be afraid to try it without as well!
Totally agree ... the u-substitution method is logically sound and promotes error-free work/steps in performing anti-differentiation ... plus, it reinforces an understanding of the four fundamental parts of a (single) integral: integrand, differential, upper limit, lower limit
Students who are starting Calculus I and beginning to know what integrals are, PLEASE just do the long process, you will better understand your problems and if you feel you grasp perfectly the patterns, then look at this video. It’s like playing chess, you don’t begin learning by applying 4 moves of tactics when you first need to understand well how the pieces moves, same with math.
Yes, I should have mentioned this is for more advanced calculus students :)
@@drpeyam perhaps if you like, you can add like a disclaimer in the description? Then the students who understand the idea can develop and get comfortable with this quicker process which saves them space and time.
this really only works when it is a linear function in the u sub since its the most basic and its the easiest way to do it mentally.
Yep
This video is mostly pointless
We learn this in England: its called integration by inspection or informally, the reverse chain rule where you consider the antiderivative, differentiate it and then adjust for the constant. However with enough practice you can do it in your head like you are 👍
Edit: We also learn u sub as a separate method.
In fact, my textbook introduced this sort of Reverse Chain Rule before introducing substitution as:
if ∫f(x)dx = F(x)+C then
∫f(ax+b)dx = 1/a F(ax+b) + C
With the other examples like 1-x^2, we had to do the substitution.
Great video btw Peyam!
I would argue this is useful in exams only. As you said in the end the problem is not using the u-sub, the problem is writing it down every time.
When still learning it is not obvious what the antiderivative is. At 3:40 for example, someone learning would have to guess between x and (1-x^2). Doing the U-sub step by step trains you in recognizing the anti-derivatives. You can do it because you have the repertoire, no one has it in the beggining of learning Calculus. All those "notice" are skills you already developped, they are the clever maths we develop while learning calculus.
So I disagree with the title. The video is still nice, don't worry.
I mention to my students that the u-substitution rule is just doing a change of basis (a topic from linear algebra) or the chain rule for integration. I think textbooks that use the term u-substitution should just drop the term and call it the chain rule for integration or more technically the change of basis rule. In any case, it is a very important rule because it is incorporated into the other techniques in math especially in calculus 2 and differential equations.
Edit: I have also seen it referred in some textbooks as the composition rule for integration just as the composition rule (chain rule for derivatives).
Very interesting
It's a little different from the change of basis (which is why some analysis texts call it change of variables), because you're considering differentiable functions rather than linear ones (different morphisms).
Great point though!
@@Awesome20801 I think you meant to type "diffeomorphisms". All good have a lovely day
I'll never forget doing integrals with my peers and being like "so what I do here is just kinda a backwards chain rule in my head" and someone said "... You mean u-sub?"
I am gifted in mathematics and still wrote down my u-subs throughout my entire Bachelor's degree. I like seeing all of my work in front of me. Not only that, but my developmental disabilities make it difficult to process information quickly and efficiently. Students should do whatever they are comfortable with and it's ridiculous to criticize the education system. Those that will benefit from not writing down will start doing it on their own.
"Students should do whatever they are comfortable with"
"it's ridiculous to criticize the education system"
How are these statements compatible when the education system doesn't let students do what they're comfortable with? The calc teachers I've met would take off points for doing integrals this way since the student isn't "showing their work".
@@martinepstein9826 Never had a single one do that, but good for you.
@@lughemblem Good for you as well.
@@martinepstein9826 I disagree and that's not true at all you are making things up
any tips on real analysis?
That's pretty much how we learned integration in the UK. You'd use substitutions for more complex stuff (generally the sub would be given) but you would do normal problems just like this.
U substitution is more standard especially for the beginners. Because it gives them a clear look to what the entire steps look like.
But having known the entire process already, it's recommended to avoid Substitution in this case.
sin(x)dx looks like d(-cos(x)), and we can move "minus" sign outside differential and integral. So we get -Integral(cos^2(x) d(cos(x))), which looks like -Integral (something^2 d(something)). Which is obviously -something^3 / 3 + C, i.e. -cos^3(x)/3 + C
I was taught this a couple weeks ago in my calc bc class, it's called "license to integrate" where you have the derivative of the inside on the outside and you can just ignore it and differentiate the inside but a much cooler name that my teacher came up with was based of the album "license to ill"
Amazing
My calculus book taught me integration like this before learning u-sub. It was called "integration by guessing".
😂
This is literally just how I was first taught to solve integrals at school, to see if you can recognise the antiderivative. It always frustrated me how I never saw anyone using it since it’s so much simpler and more intuitive.
Well it takes much more practice in normal derivatives and most students jsut dont have it so it’s more error proun for them hence why they just use a u sub which is actually cleaner at least in my opinion
U-sub can be pretty useful in some cases, for example when you want to integrate the Lambert W function
Is there a particular reason why it is called the Lambert W function? Lambert is obviously the name of someone involved in coining the term, but what does it have to do with W?
I like that you left the other attempts in the video. I do that sooo frecuently, just regret something and start all over.
My teacher called this method "GDA" (guess, differentiate, and adjust).
Cool. I tend to split the difference by writing
f(2x+1) dx = (1/2) f(2x+1) d(2x+1)
Almost as fast as the Peyam but easier to avoid mistakes.
That's literally what I've always been doing. I've always been in pain seeing my friends go through the whole process of u-sub, while I just do this (which I call "reverse chain rule") in my head. Thanks for making this video!
Well how about you do it with some proper integrals from a real calc class like int tan x^2 or ln(cos(x))^6 * x + sin x
I will write something along the lines of this...
• sin(x)cos(x)dx = sin(x)d(sin(x))
• (sin(ln|x|)/x)dx = sin(ln|x|)d(ln|x|)
I brought this up to my class in similar example. That WHEN there's a u-sub to do, often times the one's we do can easily computed directly.
My calculus teacher in high school taught us this method. It was especially useful in evaluating integrals involving the power reduction for trig squared functions
I usually like to do a shorthand u-sub:
For example, because d(-cos(x))/dx = sin(x),
cos²(x)sin(x)dx =
cos²(x)d(-cos(x)) =
d(-cos³(x)/3)
Also eliminates the fuss of having to calculate u(t_0) and u(t_1).
I never was a u sub fan in the first place lol. I just immediately put it in the equation instead of substituting u, it's more straightforward.
yep, in france we use u sub for more complicated integrals
I was always really bad at figuring out when we were given problems in which we were *supposed* to use u-sub (I simply didn't recognize if it was supposed to be needed or not) so I just usually did the problems like this, since it's a natural continuation of differentiation
In russian coursebooks there is always formula: ∫ f(k*x+n)dx = 1/k * F(k*x+n) + c, no substitution suggested for linear relatively x functions
i feel like these kind of method is useful only when you are exposed to differential equations.
sadly in our country the order of math sub for calculus as a whole are as follows:
1. differential calculus
2. integral calculus
3. differential equations
my point is.. doing U-sub is a standard thing to do.
same. in my class, if we do this, its looks like that we straightly integrate the integral without any process.
I’m glad this is something I was already taught to be familiar with. Thank you Calc II professor
I would say that this depends on your teacher. I've gotten marked down over 14% on one problem for doing the bulk of it in my head. They want to "see the process."
Interestingly, I stumbled upon this technique back in college ages ago when I ignored my professor (who simply said no) and asked the question: are there any functions other than y = e^x whose differentiation is equal to themselves? I thought about it for a long time and constructed y = (ln(x))^(x/ln(x)). This is of course pretty close to the definition of e^x but with a restricted domain, but I ended up calling the method integration by construction and was happy with the results as I learned to apply the process to other integrals. Great video!
Did you remember the formula wrong because that formula's derivative isn't equal to itself...
What about y = 0
@@Laff700 Given that ln(x)^(x/ln(x)) is literaly equal to e^x for x>0 ( by using a^b=e^(b*ln(a) ) I think you are wrong...
@@paul_w (Log[x]^(x/Log[x]))/Exp[x]≠1 though. Maybe I'm misreading it? Could you type it out in a format I could paste into Wolfram Alpha?
What you're referring to as a "restricted domain" simply means a domain for which the algebraic expression is not well defined on part of the domain. But on the portion of the domain for which it is well defined it is identical to e^x (Identity of indiscernibles). Your example is not particularly special in that one can always create a function identical to e^x but not well defined on a set (to do this for a point a, consider e^x * (x-a)/(x-a)). We normally exclude functions that are not well-defined on the expected domain precisely for that reason.
In practical settings I've only had to do an integral once. Tend to forget the rules so I still write stuff down.
Thanks for the tip! I've always thought that u-subs could be avoided, but when it came to actual problems I wasn't clear on how to go about it. Seeing it done makes the method more obvious.
So essentially this trick is intuit the antiderivative instead of applying the change of variables formula? And the argument of the video is you should be good enough at this stage to intuit what the antiderivative is
😂😂basically yes.
Leave it to this guy to give such advice
My pre Calc teacher taught us about this method and it changed EVERYTHING. Plus it was a big time saver on the Cambridge exams!
I agree. U-sub I think has a time and place but most of the time it's easier to just not use it.
My calc 2 teacher called this the 'day one shortcut' and emphasized it a lot
In Hong Kong, this method is included in most textbooks and called "Direct Integration".
As many have already stated, I have seen professors take as much as 20% off of all correct answers simply for not writing that the words of the convergent/divergent test they used even though they showed all the work required to utilize said test for the problem.
I also think it's important to always show why a particular technique used works. Writing down the steps to use during u-sub helps reinforce in my head why it works.
I know that every professor has their own particular preferences about how they want to teach, but this is the first time I've ever seen a professor show some displeasure with too much work shown? And as a tutor for calculus, I don't know how hard it has been to help a student dissect their incorrect answer when they've simply done steps in their head just to save time.
my thoughts exactly. Peyam is posting absolute doodoos nowadays
I never reflected about this and applied the substitution rule every time. Thanks 😊
Whenever I teach Calc, I make sure my students add the following rule to the list of integration rules given in the textbook, which I call the Linear Anti-Chain Rule (or, more verbosely, the chain rule for antiderivatives with linear inside functions):
Int(f’(mx+b)dx) = (1/m)f(mx+b) + C
Dr. Peyam, perhaps you can call it the Linear Anti-Chen Lu
Awesome video! I will start implementing these TODAY. I’m going to start doing integration by parts in my head as well.
What about finding an antiderivative of (e^x+xe^x)(1+ln(xe^x))? In this case, it seems natural to simplify the problem by first taking u=xe^x.
Also, the substitution method might be used later in formal derivatiions of solutions to PDE (especially when the coefficients are allowed to be distributions). Perhaps you might also use the substitution rule to cleverly rewrite a PDE prior to application of numerical methods in order to get faster convergence.
This is similar to the fact that the integration-by-parts rule has much more significance than it first appears. Without using it, I don't know how one would define the definition of a distributional derivative nor prove that it's equivalent to the standard derivative for distributions arising from functions.
*Dr. Peyam,* in the last example you could just factor out (-6)^3 out of the integrand so that you would be dealing with just (x - 2)^3 and the new constant multiplier in front of the integral. Then the antiderivative would follow smoothly.
Only really works when integrating gives something of the form k*[f(x)]^c though. U substitution is the best method for lots of integrals such as sqrt(c-x²)
Just wondering sir, where did you study math in France 🇫🇷 ? Some prépa ? (I studied Math at Ginette, Versailles). Love your Math videos BTW!
I went to the Lycée français de Vienne and the lycée français de New York
I mean… In Spain we first learn the “inspect and try to transform the integral into an immediate one” before learning any substitutions
This is so intuitive. And really it can help students recognize these things faster and simpler.
I didn’t learn integrals in my calc class yet but despite that this was so easy to understand with only knowledge of derivatives. Double integrals look pretty scary lol
As a half way point you can write it as an integrand and measure often so integral A * f * df/dx dx we see this as A/2 f^2 immediately better we can often see an adjusted chain rule directly in the integrand and again seperating the parts into the constant the chained integrand and the relative measure makes this clearer.
I would call this the jiggle method because you start with the "characteristic" answer and jiggle it about until it works for the full integrand.
when it comes to writing steps out in calculus especially, I found that everyone can do it in a different number of steps. The amount you do is determined on how well you do math in your head, trust your teacher to get it, and how much you trust yourself, which is why I write it out all the way until my hand cramps.
I think the reason why substitution is a thing is because it is an example of the pull back of differential forms under the transformation from x to U. So in case if you somehow want to study math in the future. you have already touches some examples of the pull back map
Dr P, will you be teaching differential equations or linear algebra in the spring?
Differential equations 😁
@@drpeyam I hope I get to be in your class!
Hope so too!!!
For obvious rule of chain, alright, as long as you don't mess up, but for more complicated integrals avoiding variable changes is just absurd
I always thought of situations like this as being able to use the "reverse chain rule". I wish we could do this in all scenarios though...
How about as an alternative to integration by parts. i.e. take a guess, differentiate your guess and adjust.
I go to a UK uni and my US friends would sometimes ask how I did an integral in one step instead of 2 u subs or something. Using the reverse chain rule is so much faster
I like how he flopped at 3:54 and started all over as if nothing happend😭🤣.he forgot to edit it out lmao
LMAO
LMAO
u basically asks to think chen lu in reverse.
analogically to diffrenciate a composition w/o u-sub e.g,
differentiate [(2x+1)^2], don't write f(u)=u^2, u=2x+1=>f'(u)=u^2,u'=2...
Surprised no one is mentioning you can just expand the polynomial and then easily integrate each term. No u-sub, no dividing by derivatives of linear factors. Obviously this gets trickier the higher the value of the exponent, but for linear terms that are squared or cubed, just expanding the polynomial out is very easy.
Do you think you could evaluate the antiderivative of x^(cos^2(x))?
I normally do this method in my head, but for exams and hw I do the u sub. If the answers match, I'm good!
Maybe that's something that is mainly taught in English-speaking countries. In German high schools (Oberstufe) we were taught these integrals without assigning the "rule" any name.
Right, same in French schools
Yes, integration is easy if you already know the anti-derivative.
Hi! This is the method which we learned in high school, but now in uni where we start with "proper" u-sub I've run into some troubles.
I'd like to continue doing this as it feels more intuitive, but I've noticed that some integrals are just impossible for me to solve "in my head" since I cannot only multiply/divide by a constant in order to achieve the inner derivative in my integrand. For example:
e^sqrt(x)
I understand that the inner derivative is 1/2 * x^(-1/2), but since I don't have any factor of x in my original integrand I don't understand how this is possible to solve without the "proper" u-sub.
Moreover, I don't understand why it works with proper u-sub if u-sub truly just is the reverse chain rule.
Any help would be appreciated!
For this you have to use u sub, I was referring to the easier cases
@@drpeyam Thank you for the answer!
My university courses are filled trash linear algebra and pointless abstract algebra i wish there was a much greater emphasis on integral calc
I thought in an exam you to show your steps and the answers may only count as 1 or 2 Marks, whilst the methodology counts for most. If I was marking you I would deduct marks for taking shortcuts.
I was expecting you to say, "don't u-sub, just undo the CHEN LU"
Is that the Blackpen/Redpen style of saying "chain rule"?
gf: Are you watching Mr. Rogers?
Me: Better
Shouldn't this 'U substitution' just be called reverse chain rule or chain rule for integrals, given that it is usually used for 'nested' functions like log( (3x + 4)^8 ) ?
Dr Peyam please do more complicated Integration without substitution method.
This is what will hit students pretty quick in calc 2. That's when I said there's no time for a measly u-sub, it's time to integrate.
Dear Prof. Payam, the problems are two: 1 - it is not teached to memorize generilized immediate integrals; 2 - no model is taught, because of using memory requests too effort.
The conclusion is that student think that they can study math without memorizing, which is impossible.
Greetings from Italy.
True in France we only change variables when needed tbh. At least I do.
We're taught to recognise derivatives like he does in the video
Love watching a master showing off his craftmenship
You can lead a student to knowledge, but you can't make them think.
Any student can memorize a method, but some just don't have the intelligence to use these kinds of shortcuts.
Can I use this technique if there are a (e and ln) function in the math problems???
What surprises me is that even among the good students who use the u-sub correctly in the more sophisticated integrals (more like Calc 2 stuff), don't really know WHY the u-sub actually works. And when digging into the textbooks, apart from a formal proof (typically in some appendix) there is no proper (geometrical) explanation as to why it actually works. Sadly...because in essence the u-sub is a "reverse" operation from the chain rule. At least, that is how it could be introduced in Calc 1. But most students learn the u-sub as a "step" problem step 1,2,3 and done.
Excellent comment, I completely agree! They completely use it as a black box. I think that’s why in Calc 3 they still write down all the steps
I know some markers who would dock marks because 'You didn't show all the steps'.
If you are needing to demonstrate to someone else that you know what you are doing it makes sense for them to ask you to show all steps. When doing it for yourself of course you don't need to.
i learned this as 'reverse chain rule', separate from u sub.
No. I am good at calculus. I'm studying for my multivariable calculus final exam right now actually.
BUT when the problems are more difficult than the simple stuff here and the number of steps and processes to get your final answer increase, it takes more time to write out your steps but saves in the time you would spend double tracking to check why your answer is wrong. And for an exam, you don't have that luxury. If you're wrong you're wrong, but now you don't get partial credit.
I also tutor in calculus by the way and I NEVER recommend skipping steps like that. I sometimes do, but only when I don't have any doubt about the result. People still learning should not be doing things like that, because it will just make it harder.
So the reason I’m mentioning this is because some of my multivariable students spend 5 mins writing out a simple u sub, which makes them lose valuable time, especially if they have to evaluate 4 double/triple integrals, that’s why I recommend them to do it mentally
@@drpeyam I get that, but from the students perspective it is way too hard to check my work when I get done doing all those integrals. It's too much to keep in my head, even tho I've had a ton of practice. Sometimes I can do it but most of the time I can't trust myself with something as high stakes as an exam. One mistake is all it takes. For me the most important thing is book-keeping.
After learning about the reverse chain rule in my calc class after self learning U sub from calc channels, I noticed that U sub is a cumbersome crutch. However, the related skills of U sub do carry over towards trig substitutions, but that is the only one I could think of that could be useful.
I'm french and this is so true
I like trying to do the steps implicitly in my head, but sometimes the Algebra ends up tripping me up. Like I'll forget a constant factor of pi or something because it pops out of the Algebra about the du, or I'll miss a sign or mis-multiply when evaluating the integral.
Since we are not electronic computers, we shouldn't make becoming one a goal.
:-) Dear colleague Peyam - doing multiple steps at the same time in your mind - it might create mistakes (divide or multiply by 2/3 ??). So, if I want to be sure that I'm correct, even I prefer to insert an extra step (may be I'm a bit prone to stupid mistakes)
Linear functions are so nice and that should be taken advantage of whenever possible
Peyam Method = Blah substitution, where Blah = Estimate then Repair
Did you misspeak when you said "multivariable exam?" That question is clearly from Calc 1 or Calc 2.
Also, the method you used is basically how kids are taught that they need U-sub. PLUS U-sub, the method not only is usful for this, but also get's students used to using the letters U (sometimes V too) so when they do IBP, they'll be used to seeing UV-int(vdu).
No I didn’t misspeak, I literally had some Calc 3 students spend 5 mins on a u sub, which inspired me to make this video
It doesn't sound very rigorous. It sounds like just pulling the answer out of nowhere by magic and looks like the questions are contrived to work well with the method.