It doesn’t matter if it’s ILATE or LIATE. Since if you have an integral with both log and inverse function, then it’s most likely not doable in the first place 😆
I agree in not using LIATE, it's not about memorising a mechanic, I like having the openness of realising a mechanic doesn't always work. When I teach integration by parts, I always teach find the function to integrate first
I believe that by inspection we can easily see which one is easier to differentiate and which one to integrate. This idea or sense helps students in the long run. Tricks might help for short term, but not for the long run. This also helps students to recognise different patterns and get familiar with mathematics.
AMEN. Preach it. Ok, but seriously, these are the exact same words I had in mind as soon as I started the video, and I am glad I am not the only one, and I am glad you beat me to it.
Define formally and mathematically what "easier" to integrate or differentiate means. Otherwise it is bringing some pseudoscience argument into the mix. As far as I can tell there are dozens of patterns which work by trial and error, and a few of them are very common. But the patterns should ultimately be well defined and not rely on intuition. A computer program should be fed thousands of these problems applying pattern rules until a "perfect" pattern is found at least for those in the dataset
@@gregorymorse8423 The patterns need not be well-defined. LIATE is merely mnemonic, not a mathematical theorem. You are reading too much into this, and nearly everyone else knows what exactly is being meant by "easier".
I feel like most Mathematicians do this anyways subconsciously. And it is a good first instinct to have to quickly solve problems. But there are cases that don't quite work, and I think it's up to the student to discern for themselves.
My calc 2 teacher taught us the LIATE method but didn't make us use it if we didn't want to. Same with the DI method. I personally never used it but for some people it did help. IMO it's just about teaching the tools, not telling the students which ones to use.
Yeah same. Looking back at it, I think it was better that I never used methods like LIATE as this forced me to think more about which function to integrate and which to differentiate when using integration by parts. After some practice, I could calculate the second term quickly without a pen often and I could see if there was any way to integrate that term or if it was similar somehow to the original integral (like in case of sec³(x)) or if I had to do repeated integrations by parts which were getting even more complicated etc.
Another way to deal with sec^3 is: sec^3(x)= 1/cos^3(x)= cos(x)/cos^4(x)= cos(x)/(cos^2(x))^2=cos(x)/(1-sin^2(x))^2 and with a substitution: sin(x)=u ; cos(x)dx=du we have the integral of 1/(1-u^2)^2 wich is a rational funcion, so quite easy to integrate. Is a method that works for 1/cos^n(x)and 1/sin^n(x) for every odd n.
This video is great. There's always that tension between "tools that are very helpful most of the time" and "concepts that work nearly all the time," and you balanced this excellently in this video with examples to both sides. 💪
Excellent video. Before I retired, I told my students that LIATE is a nice "rule of thumb" but, when integration by parts is applicable, LIATE does not work 100% of the time. The example I used to demonstrate this fact is the integral of (xe^x)/(x+1)^2. in this case, the factor to be differentiated is xe^x and the factor to be integrated is 1/(x+1)^2.
My first class of calculus was in 11th grade in India, that's Junior Year in America. We were always taught ILATE, explanation was "Choose whichever is the harder to integrate" and like you said, it works in almost all scenarios. Wherever there was an exception we were given the solution and were told the specifics. It's the first time I'm hearing of DI method. Pretty amazing!!
I never seen the LIATE method before. Like you said, I tell the students that they must figure out what to integrate first since finding the derivative is the easiest (sometimes). Unfortunately, many students of mine have trouble doing so.
The way I learned the LIATE method was that you should use it when first attempting the question. They said it won't always work, but it can help you start working out the problem.
Integration by parts: int udv=uv-int vdu One day i see a cow doing i dont know what.... a engineer cow with uniform xd "Un Día Vi a Una Vaca Vestida De Uniforme"
I never thought LIATE as a rule or as a method. It's more like an advice from experienced mathematicians, "In most of the integrations, you'll find yourself integrating in this pattern."
I am lucky that I got a good Mathematics Teacher . When we came across this stuff , he taught us this method but suggested not to use it . He told us to integrate by parts by choosing which expression is "easier to integrate" by our intuition. And for real , in some questions even this rule fails(as in takes longer time ). And you sometimes can use different methods to integrate a function that seems harder to integrate but is easier than the other one.
For integral of xsin(x^2)dx in the end card: its easier to do u-sub then by parts u=x^2 , dx=du/2x integral of xsin(x^2)dx =1/2 * integral of sin(u)du =1/2 * -cos(u) =-1/2 * cos(x^2)+c
For the integral of xsin(x^2) there is no need of integration by parts. Just U-substitution let u=x^2 so du=2xdx It becomes integral of-1/2cos(u) and the final answer is-1/2cos(x^2)+C
Yep, and in fact, if you do LIATE integration by parts on this integral, then you'll have to differentiate x and integrate sin(x^2), the latter of which is impossible!
My teacher didn’t like the idea of restricting us to LIATE so we instead subscribed to a some rules of thumb. These are the ones I remember, still use, and are almost always enough to make the right choice. A) If one of the functions has cyclic (sines, cosines, exponentials, etc.) or terminal derivatives (polynomials), let that one be the one you differentiate. Unless the other function’s integral is unknown or unsolvable by you, in which case… B) Let the most difficult of the functions that you CAN use integrate be the one you integrate.
It's fun, in France we have ALPES, for arccos/arcsin ; log ; polynomial ; exponential ; sinusoidal. At the time I learnt integration, my teacher said like you that it is better to search which part is better to integrate than use this tip.
In spanish we say ALPES: A=arcsin, arccos, arctan, etc. L=logs P=polynomios E=exponential S=sin, cos, tan, etc. It's said that not always work but it has always work for me with this set.
In my college, they had taught us about the ILATE rule but they also mentioned that it is not always necessary to follow the order and gave us the examples. So we just had to understand which function was easily integrable and which wasn't. I guess it is more sort of, an intuitive method of solving.
Let me tell you another approach to the same problem we assume lnx=t dx=e^xdx integral reduces to e^t[ t/(t+1)²]dt e^t [ 1/1+t - 1/(1+t)²]dt this becomes d(e^t/1+t) using the product rule and the derivative and integral cancel out. so we get the same answer substituting t back
for guys lived in morocco 🇲🇦, they have been using a technic called "ALPES" istead of laatte A -Arctan ,arcsin,arcos L - ln P - polynomial E - Exponential S -Sin , cos , tan
I completely agree, you can't exactly impose strict rules for calculus. It always matters the type of problems you are solving and thinking a few steps ahead is the key. I always hated that rule
Me too. Unfortunately, general chemistry is riddled with them. The entire class is about learning a bag of tricks and rules to solve contrived problems from every different area of chemistry. There's no natural structure or flow to the class. For example, in a calc class, you'll typically start with a motivation for derivatives, then discuss the definition of a derivative, then learn how to arrive at derivative "rules" by using what you already know, then repeat for integrals. It flows very nicely, and there's a general structure to the class. Not so in gen chem. You start with the absolute basics, then learn stoichiometry for two weeks, then some organic chemistry for the next two weeks, then nuclear chemistry for another two weeks, etc. The lack of flow in such a class necessitates the use of black-box tricks and rules to complete the problems. In short, you never learn where the tricks come from, why they work, or how to work without them, because you never go far enough into a subject to be able to understand such things. This is why general chemistry is so badly done across the board, the nature of the class attracts people who are drawn to arbitrary rules and control (the types who tell you to follow a procedure without any explanation for WHY you should follow the procedure). I despise any class that teaches an arbitrary set of tricks and rules instead of the skills necessary to develop your own tools.
You can thank high school math for those "tricks". Heck, most AP calculus classes do not even have the courtesy to teach epsilon and delta proofs. Not including the precise definition of a limit is the biggest detriment of high school calculus because it's the heart of college level calculus and analysis where such arguments are the foundations for proving derivatives and integrals. If want to know why the definition of a derivative is the limit of a difference quotient you need epsilon delta arguments and likewise for an integral as a limit of a riemmann sum.
Looks like you don't know how to cook them: first show trick than explain why it works. There are no math olympiad winners who dont know endless row of tricks. Yes, they are clever but not " invent method of general sulution of a*cosx+b*sinx = c in 1 second"-clever.
I tried to use LIATE in the bonus example. So I chose sin(x^2) to be integrated. When I tried to do that found out it becomes soooo easy if you just use the form 1/2∫2xsin(x^2)dx.
Moral of the story is just do whatever sounds like less of a nightmare and fuck around if you’re stuck. Liate is just a way to get you moving if you’re a hesitant student. Only way to solve a hard problem is to try solving it, even if there’s no clear solution.
It’s actually funny that in my education in Dominican Republic we use the method but the order is different, for us is ILATE, all the letters stands for the same.
In the bonus example, just substitute u=x^2 I know the question is what if I try to integrate by parts but: - maths is about getting to the right answer and spotting the quickest way is always a valuable skill - according to the rules of logic ("if today's Sunday I'm the pope" according to the late Doctor Richard Maunder, or "if my granny had wheels she's be a bike" according to Gino Cappucino), any answer to your question is correct if I'm not going to integrate by parts. If I integrate that by parts then I'll climb Nelson's Column naked.
I never heard of this rule of thump but the use of sec disturbs me very much. Here in germany I never saw someone using sec, csc or cot, they just use the normal sin, cos, cot representation.
I don't mention LIATE. Let students explore on their own. If their u fails, then go back and try a different u. My tip for students is to find the u-term first. The u-term in the original integrand is usually the term you want to change or get rid of through differentiation.
I never heard of this LIATE method and maybe it was for the best. Instead of giving us another "trick" to memorize, they just told us to choose which to integrate and which to differentiate.
I often tell my students, "We artificially adjust things to fit into standard forms we can use." Especially with situations we are integrating constant/quadratic or linear/quadratic.
We indians learned ILATE rule this also we don't use maximum cases We apply integration by parts By just checking which is more preferable to integrate
I have never taught that LIATE method and I get slightly annoyed when students ask me about it. That is because integration by parts is SUPPOSED to be difficult, you are supposed to struggle to make that choice because the struggle is how you learn and become gradually comfortable with the concepts. If it becomes just an algorithm to memorize and apply, then there is literally no point to studying this, you might as well just ask Photomath to do your integrals for you because you are not learning anything.
Never heard of LIATE. Though for a sec that was an weird English abbreviation for “Integrating by parts” because I didn’t learn math in English (also never had an equivalent to LIATE taught to me). So I thought this video was about why you didn’t teach integrating by parts. Lmao
I feel like another way of integration people should start utilizing more often is identifying forms of derivatives within the integrand. Whenever I can, I prefer to skip using some sort of U-sub or By Parts simply because I have the opportunity to use algebraic manipulation to force the integral in the form of a derivative. With the integral of lnx/(1+lnx)^2, I couldn't help but to think it may be some quotient rule in disguise, so you can end up rewriting the numerator as 1*(1+lnx) - x*(1/x), 1 being the derivative of x and 1/x being the derivative of (1+lnx), so from there you can identify that its the derivative of x/(1+lnx) by the quotient rule.
4:49 we just took this in class today and a student kept integrating by parts till he had about 10 terms because he didn't add the integration of sec^3 x on both sides 😂
One thing that I like about studying the later math courses in university is that the teachers for those courses don't care how I solve an integral - they are basically like "just solve this integral in some way", and only care about the answer. I am even allowed to use Wolfram-Alpha for example problems during classes if I feel like it, lol.
That's pretty much what you'd do in the real world anyway. Unless you are competing in MIT's integration bee, or discovering new mathematics, chances are, you'll have access to Wolfram Alpha in any situation outside an academic exercise, where you'd need to compute an integral. The reason teachers who initially teach the method, want you to use the method they teach, is to make it easy to grade. Once it's no longer the main substance of the topic you're working with, no one will care what specific method you choose. In computer science, you might need to know how to strategically pick the method that is least computationally intensive, to make your program more efficient, when it will have to do hundreds if not thousands of repeats of it, but that's the only time I can see a right or wrong choice of method, when faced with two options that get you the same solution.
OMG the integral at 9:05 I just didn't solve it with LIATE I solved it using partial fractions can't believe it 😋😋😋😋 wish u can see my solution it gave the final answer directly 😂😂
There are two ways for this triple product of x, sin(x), and e^x. One method is to use complex numbers, to rewrite sin(x)*e^x as a sum of two complex exponential functions, which will be i/2*e^([1 - i]*x) - i/2*e^([1 + i]*x). Pull the i/2 out in front as a constant, and assign constants N = 1 - i and P = 1 + i, to simplify our writing. Now you can integrate i/2*x*[e^(N*x) - e^(P*x)], with only one IBP table, after pulling i/2 out in front as a constant, and it is a simple ender. Then convert it back to the real world. Another way you can do it, is to assign J and K such that: J = integral sin(x)*e^x dx, and K = integral cos(x)*e^x dx Suppose we've previously solved these two integrals. The results are: J = 1/2*[sin(x) - cos(x)]*e^x K = 1/2*[sin(x) + cos(x)]*e^x We'll use J & K, as we construct our IBP table. Let sin(x)*e^x be integrated, and x be differentiated: S _ _ D _ _ I + _ _ x _ _ sin(x)*e^x - _ _ 1 _ _ J + _ _ 0 _ _ integral J dx Construct our result: x*J - integral J dx Since J is a linear combination of sin(x) and cos(x), this means integral J dx is a linear combination of J & K: integral J dx = 1/2*J - 1/2*K Thus, our result is: x*J - [1/2*J - 1/2*K] = (x - 1/2)*J + K/2 Fill in J & K: (x - 1/2)*1/2*[sin(x) - cos(x)]*e^x + 1/4*[sin(x) + cos(x)]*e^x Consolidate and simplify, and add +C: 1/2*[x*sin(x) - x*cos(x) + cos(x)]*e^x + C
I mean ILATE is easier to memorize and teach to students and u can identify the integral faster rather than trying to find which one can be integrated. Also integrals having both as same function are rarely seen in school level math at least in my schools
Well for the integral(ln(x)/(1+ln(x))^2 ,x) you can solve this with just u sub and integration by parts using the LIATE method. No need for creativity on that one Let u = 1+ln(x) integral( ln(e^(u-1))*e^(u-1)/u^2 ,u) this simplifies and expands to integral(e^(u-1)/u ,u)-integral(e^(u-1)/u^2 ,u) then integral(e^(u-1)/u^2 ,u) use integration by parts by integrating 1/u^2 turning into -e^(u-1)/u+integral(e^(u-1)/u ,u) this means the integral(e^(u-1)/u ,u)-integral(e^(u-1)/u^2 ,u)=integral(e^(u-1)/u ,u)-( -e^(u-1)/u+integral(e^(u-1)/u ,u)) that simplifies to e^(u-1)/u putting x back you get the x/(1+ln(x))
@@chanuldandeniya9120 Expand the expression (u-1)*e^(u-1)/u^2 into two fractions and you get e^(u-1)/u-e^(u-1)/u^2. One of these fractions can be integrated separately as I have shown above using Integration by parts. I should have made the step where I separated them more clear but yes what you have is an intermediate step. I guess what makes my solution creative is that the integrals cancel out rather than being fully computed.
For most of our IBP problems, when we have a transcendental function and a power function, if the derivative of the transcendental function becomes algebraic, then it is u, if it stays transcendental, then it is with dv.
@@kepler4192 ok. differentiate sin(cos(tan(csc(sec(cot(sinh(cosh(tanh(csch(sech(coth(arcsin(arccos(arctan(arccsc(arcsec(arccot(arcsinh(arccosh(arctanh(arccsch(arcsech(arccoth(x))))))))))))))))))))))))
It doesn’t matter if it’s ILATE or LIATE. Since if you have an integral with both log and inverse function, then it’s most likely not doable in the first place 😆
Yup 👍
What's your qualifications
Are you a PhD holder?
In india it is called ilate 🤔🤔
@@AbhishekKumar-jg7gq ya..you ar right..
@@AbhishekKumar-jg7gq mere ko LIATE sikhaya hai lol
I prefer the LATTE method. If the integral looks hard, go and make yourself a coffee.
Haha good one 😂
XD
😂😂😂😂😂😂😂😂😂😂😂😂😂👏
Noice
Gigantic heart palpitations, here we go!!!!!!!
12:50
The smile of cancellation
best cancelling i've seen in a while
0 and 1 are so beautiful!
I agree in not using LIATE, it's not about memorising a mechanic, I like having the openness of realising a mechanic doesn't always work. When I teach integration by parts, I always teach find the function to integrate first
I believe that by inspection we can easily see which one is easier to differentiate and which one to integrate. This idea or sense helps students in the long run. Tricks might help for short term, but not for the long run. This also helps students to recognise different patterns and get familiar with mathematics.
I agree. It’s better to understand why you shouldn’t do a certain thing, instead of just learning a rule to say you shouldn’t do it.
AMEN. Preach it. Ok, but seriously, these are the exact same words I had in mind as soon as I started the video, and I am glad I am not the only one, and I am glad you beat me to it.
Yes this helps a lot in integrating products of similar functions. Like in sec³(x), one can easily say that we will integrate sec^2(x)
Define formally and mathematically what "easier" to integrate or differentiate means. Otherwise it is bringing some pseudoscience argument into the mix. As far as I can tell there are dozens of patterns which work by trial and error, and a few of them are very common. But the patterns should ultimately be well defined and not rely on intuition. A computer program should be fed thousands of these problems applying pattern rules until a "perfect" pattern is found at least for those in the dataset
@@gregorymorse8423 The patterns need not be well-defined. LIATE is merely mnemonic, not a mathematical theorem. You are reading too much into this, and nearly everyone else knows what exactly is being meant by "easier".
I feel like most Mathematicians do this anyways subconsciously. And it is a good first instinct to have to quickly solve problems. But there are cases that don't quite work, and I think it's up to the student to discern for themselves.
I agree with you of doing these subconsciously.
My calc 2 teacher taught us the LIATE method but didn't make us use it if we didn't want to. Same with the DI method. I personally never used it but for some people it did help. IMO it's just about teaching the tools, not telling the students which ones to use.
Yeah same. Looking back at it, I think it was better that I never used methods like LIATE as this forced me to think more about which function to integrate and which to differentiate when using integration by parts. After some practice, I could calculate the second term quickly without a pen often and I could see if there was any way to integrate that term or if it was similar somehow to the original integral (like in case of sec³(x)) or if I had to do repeated integrations by parts which were getting even more complicated etc.
I had a question?
Is there like a proof to the ilate rule
Another way to deal with sec^3 is:
sec^3(x)= 1/cos^3(x)= cos(x)/cos^4(x)= cos(x)/(cos^2(x))^2=cos(x)/(1-sin^2(x))^2
and with a substitution: sin(x)=u ; cos(x)dx=du
we have the integral of 1/(1-u^2)^2 wich is a rational funcion, so quite easy to integrate.
Is a method that works for 1/cos^n(x)and 1/sin^n(x) for every odd n.
But is that really simpler?
∫ sec³x dx = ∫ 1/cos³x dx = ∫ cos x/cos⁴x dx = ∫ cos x/(1−sin²x)² dx → sin x = u, cos x dx = du
= ∫ 1/(1−u²)² du = 1/4 ∫ ( 1/(1+u)² + 1/(1−u)² + 1/(1+u) + 1/(1−u) ] du
= 1/4 ( −1/(1+u) + 1/(1−u) + ln|1+u| − ln|1−u| ) + C
= 1/4 ( 2u/(1−u²) + ln|(1+u)/(1−u)| ) + C
= 1/2 ( sinx /(1−sin²x) + 1/2 ln|(1+sinx)/(1−sinx)| ) + C
= 1/2 ( sinx/cos²x + 1/2 ln|(1+sinx)²/(1−sin²x)| ) + C
= 1/2 ( secx tanx + 1/2 ln|(1+sinx)²/cos²x| ) + C
= 1/2 ( secx tanx + ln|(1+sinx)/cosx| ) + C
= 1/2 ( secx tanx + ln|secx + tanx| ) + C
The solution above doesn't even include the work required to find partial fraction decomposition.
@@MarieAnne.Hey who are you.
.
.
.
.
you really an god gifted child.
This video is great. There's always that tension between "tools that are very helpful most of the time" and "concepts that work nearly all the time," and you balanced this excellently in this video with examples to both sides. 💪
Excellent video. Before I retired, I told my students that LIATE is a nice "rule of thumb" but, when integration by parts is applicable, LIATE does not work 100% of the time. The example I used to demonstrate this fact is the integral of (xe^x)/(x+1)^2. in this case, the factor to be differentiated is xe^x and the factor to be integrated is 1/(x+1)^2.
I think the answer is e^x/(x+1)+c
@@clementfradin5391 You are correct!
we said U times V - the integral of V du. ... How math geeks make poems
Thanks @richardryan5826 ! This problem helped me!
My first class of calculus was in 11th grade in India, that's Junior Year in America. We were always taught ILATE, explanation was "Choose whichever is the harder to integrate" and like you said, it works in almost all scenarios.
Wherever there was an exception we were given the solution and were told the specifics.
It's the first time I'm hearing of DI method. Pretty amazing!!
By parts ke kaam asan ho jaata hai...
Hame
LATEC padhaya Gaya hai
Small brain: memorize this rule
Big brain: recognize the easier integral like a boss
I never seen the LIATE method before. Like you said, I tell the students that they must figure out what to integrate first since finding the derivative is the easiest (sometimes). Unfortunately, many students of mine have trouble doing so.
The way I learned the LIATE method was that you should use it when first attempting the question. They said it won't always work, but it can help you start working out the problem.
Integration by parts:
int udv=uv-int vdu
One day i see a cow doing i dont know what.... a engineer cow with uniform xd
"Un Día Vi a Una Vaca Vestida De Uniforme"
wtf xd
@@cookieman2028 That says "one day I see a cow dressed in a uniform"
I never thought LIATE as a rule or as a method. It's more like an advice from experienced mathematicians, "In most of the integrations, you'll find yourself integrating in this pattern."
For the past 3 years before I started my own RUclips channel, I ever called you Father of Calculus 💪
-cos(x^2)/2+C will be the answer to the last question. No need to apply LIATE or ILATE. It can easily be done by substitution by substituting x^2=t.
K genieass
Whenever i see x^2 and x together , always do it
@@Katoto112 why do you even care lol
Integration by parts felt impossible until i found this video, thanks!
I am lucky that I got a good Mathematics Teacher . When we came across this stuff , he taught us this method but suggested not to use it . He told us to integrate by parts by choosing which expression is "easier to integrate" by our intuition. And for real , in some questions even this rule fails(as in takes longer time ). And you sometimes can use different methods to integrate a function that seems harder to integrate but is easier than the other one.
The last line was a twister lmao.
Very insightful. I like the DI method for IBP.
For integral of xsin(x^2)dx in the end card: its easier to do u-sub then by parts
u=x^2 , dx=du/2x
integral of xsin(x^2)dx
=1/2 * integral of sin(u)du
=1/2 * -cos(u)
=-1/2 * cos(x^2)+c
dont forget the C 😅
welldone, it's a right ans. I got it too
@@thekingdragon6660 damn how did i forget the c 😰
Thanks lol
We learned "choose one, and check if you've reduced the mess. Back out if you have not." Would a substitution on u = lnx not help on the 4th case?
For the integral of xsin(x^2) there is no need of integration by parts. Just U-substitution let u=x^2 so du=2xdx
It becomes integral of-1/2cos(u) and the final answer is-1/2cos(x^2)+C
Yep, and in fact, if you do LIATE integration by parts on this integral, then you'll have to differentiate x and integrate sin(x^2), the latter of which is impossible!
In my 6 years of knowing integral calculus this is my first time that I've heard of LIATE. At this point I would trust my intuition more than LIATE.
For the bonus problem, i think we can use u substitution u=x^2
Well... I've been taught ILATE as integration by parts
Doesn't matter
@@nikhilnagaria2672 It does
Its the same thing. L and I both have same level of importance.
@@marco-vz5kv how?
@@magnumfang yeah
when you put +C it makes me happy
The last bonus question can be done by substitution
The third integral came in handy when i was computing distance traveled by a projectile. Glad I did it right
My teacher didn’t like the idea of restricting us to LIATE so we instead subscribed to a some rules of thumb. These are the ones I remember, still use, and are almost always enough to make the right choice.
A) If one of the functions has cyclic (sines, cosines, exponentials, etc.) or terminal derivatives (polynomials), let that one be the one you differentiate. Unless the other function’s integral is unknown or unsolvable by you, in which case…
B) Let the most difficult of the functions that you CAN use integrate be the one you integrate.
It's fun, in France we have ALPES, for arccos/arcsin ; log ; polynomial ; exponential ; sinusoidal. At the time I learnt integration, my teacher said like you that it is better to search which part is better to integrate than use this tip.
Wow that's such a cool acronym!
Well I guess this acronym really… alp-ed you out!
@@goodplacetostop2973 lol
@@goodplacetostop2973 you’re a celebrity on this part of RUclips.
In Mexico we use ALPES too and it's dope
for the sex^3(x) qn we can use the result of int root(a^2 + x^2) dx = x/2 root(1+x^2) + a^2/2 ln |x + root(1+x^2)| + c to get quick ans
In spanish we say ALPES:
A=arcsin, arccos, arctan, etc.
L=logs
P=polynomios
E=exponential
S=sin, cos, tan, etc.
It's said that not always work but it has always work for me with this set.
5:23 actually it helps u need to know an identity of integration of root of x^2 + a^2
6:24 secx=under root(1+tan²x)
So eqn becomes underroot(1+tan²x)×sec²x
Let t=tanx then subsitution you also get answer
In my college, they had taught us about the ILATE rule but they also mentioned that it is not always necessary to follow the order and gave us the examples. So we just had to understand which function was easily integrable and which wasn't. I guess it is more sort of, an intuitive method of solving.
The feeling being ready before diving to Calculus 2 this upcoming 2nd sem.😌
The 4th is amazing !
Let me tell you another approach to the same problem
we assume lnx=t
dx=e^xdx
integral reduces to e^t[ t/(t+1)²]dt
e^t [ 1/1+t - 1/(1+t)²]dt
this becomes d(e^t/1+t) using the product rule and the derivative and integral cancel out.
so we get the same answer substituting t back
They way he solved the hard integral blew my mind! I've never heard of LIATE, so this entire video is just me learning.
Nice!
for guys lived in morocco 🇲🇦, they have been using a technic called "ALPES" istead of laatte
A -Arctan ,arcsin,arcos
L - ln
P - polynomial
E - Exponential
S -Sin , cos , tan
the same as ILATE
Thank you!
I completely agree, you can't exactly impose strict rules for calculus. It always matters the type of problems you are solving and thinking a few steps ahead is the key. I always hated that rule
I personally hate those sort of "tricks". Make the students think by themselves.
Me too. Unfortunately, general chemistry is riddled with them. The entire class is about learning a bag of tricks and rules to solve contrived problems from every different area of chemistry. There's no natural structure or flow to the class. For example, in a calc class, you'll typically start with a motivation for derivatives, then discuss the definition of a derivative, then learn how to arrive at derivative "rules" by using what you already know, then repeat for integrals. It flows very nicely, and there's a general structure to the class. Not so in gen chem. You start with the absolute basics, then learn stoichiometry for two weeks, then some organic chemistry for the next two weeks, then nuclear chemistry for another two weeks, etc. The lack of flow in such a class necessitates the use of black-box tricks and rules to complete the problems. In short, you never learn where the tricks come from, why they work, or how to work without them, because you never go far enough into a subject to be able to understand such things. This is why general chemistry is so badly done across the board, the nature of the class attracts people who are drawn to arbitrary rules and control (the types who tell you to follow a procedure without any explanation for WHY you should follow the procedure). I despise any class that teaches an arbitrary set of tricks and rules instead of the skills necessary to develop your own tools.
You can thank high school math for those "tricks". Heck, most AP calculus classes do not even have the courtesy to teach epsilon and delta proofs. Not including the precise definition of a limit is the biggest detriment of high school calculus because it's the heart of college level calculus and analysis where such arguments are the foundations for proving derivatives and integrals. If want to know why the definition of a derivative is the limit of a difference quotient you need epsilon delta arguments and likewise for an integral as a limit of a riemmann sum.
Looks like you don't know how to cook them: first show trick than explain why it works. There are no math olympiad winners who dont know endless row of tricks. Yes, they are clever but not " invent method of general sulution of a*cosx+b*sinx = c in 1 second"-clever.
The bonus question let u=x^2 and you have to solve 1/2∫sin(u)du
I tried to use LIATE in the bonus example. So I chose sin(x^2) to be integrated. When I tried to do that found out it becomes soooo easy if you just use the form 1/2∫2xsin(x^2)dx.
I haven't finished the video, but I'm so much satisfied by the way you change markers (so Smooth dude) 😂❤️
Moral of the story is just do whatever sounds like less of a nightmare and fuck around if you’re stuck. Liate is just a way to get you moving if you’re a hesitant student. Only way to solve a hard problem is to try solving it, even if there’s no clear solution.
I usually write
Int x^2 ln(x) dx = Int ln(x) d(x^3/3) = ln(x) x^3/3 - Int x^3/3 d(ln(x))= ln(x) x^3/3 - Int x^2/3 dx = ln(x) x^3/3 - x^3/9 + C
It’s actually funny that in my education in Dominican Republic we use the method but the order is different, for us is ILATE, all the letters stands for the same.
In the bonus example, just substitute u=x^2
I know the question is what if I try to integrate by parts but:
- maths is about getting to the right answer and spotting the quickest way is always a valuable skill
- according to the rules of logic ("if today's Sunday I'm the pope" according to the late Doctor Richard Maunder, or "if my granny had wheels she's be a bike" according to Gino Cappucino), any answer to your question is correct if I'm not going to integrate by parts. If I integrate that by parts then I'll climb Nelson's Column naked.
Scandalous! I just was taught this on Monday!
I never heard of this rule of thump but the use of sec disturbs me very much. Here in germany I never saw someone using sec, csc or cot, they just use the normal sin, cos, cot representation.
I don't mention LIATE. Let students explore on their own. If their u fails, then go back and try a different u. My tip for students is to find the u-term first. The u-term in the original integrand is usually the term you want to change or get rid of through differentiation.
The last screen integral is u-sub, just let u=x² and we it autopilots on itself.
I never heard of this LIATE method and maybe it was for the best. Instead of giving us another "trick" to memorize, they just told us to choose which to integrate and which to differentiate.
YAY ! Just took membership ! (both channels)
In India we have ILATE ....
I often tell my students, "We artificially adjust things to fit into standard forms we can use."
Especially with situations we are integrating constant/quadratic or linear/quadratic.
the virgin LIATE vs the chad Tabular integration vs the gigachad bessel function
In Malaysia syllabus we used LPET , logarithm, polynomial, exponential and trigo.
We indians learned ILATE rule
this also we don't use maximum cases
We apply integration by parts
By just checking which is more preferable to integrate
I have never taught that LIATE method and I get slightly annoyed when students ask me about it. That is because integration by parts is SUPPOSED to be difficult, you are supposed to struggle to make that choice because the struggle is how you learn and become gradually comfortable with the concepts. If it becomes just an algorithm to memorize and apply, then there is literally no point to studying this, you might as well just ask Photomath to do your integrals for you because you are not learning anything.
The way i teach it, its usually usable if there is no composite function involved in the integrand.
Never heard of LIATE. Though for a sec that was an weird English abbreviation for “Integrating by parts” because I didn’t learn math in English (also never had an equivalent to LIATE taught to me).
So I thought this video was about why you didn’t teach integrating by parts. Lmao
sir u look like a "LAOXIAN"- for those who don't know(laoxian means oldmonk with crazy abilities.)
he's morphing to an ancient philosopher video by video lol
Another (UK) who hasn't seen this before
As I understand it, it helps in a lot of occasions, but you can't always see when it *won't* work
It's not LIATE IT'S ILATE
liate is typical in this question . So,Without using liate answer is comming --(1/2) cos(x^2 ) +c
I feel like another way of integration people should start utilizing more often is identifying forms of derivatives within the integrand. Whenever I can, I prefer to skip using some sort of U-sub or By Parts simply because I have the opportunity to use algebraic manipulation to force the integral in the form of a derivative. With the integral of lnx/(1+lnx)^2, I couldn't help but to think it may be some quotient rule in disguise, so you can end up rewriting the numerator as 1*(1+lnx) - x*(1/x), 1 being the derivative of x and 1/x being the derivative of (1+lnx), so from there you can identify that its the derivative of x/(1+lnx) by the quotient rule.
Off by a minus sign on that last integral.
My students call LIATE Larbage.
We shall call your method the Beach Boys method, because... "Wouldn't it be nice?" :)
4:49 we just took this in class today and a student kept integrating by parts till he had about 10 terms because he didn't add the integration of sec^3 x on both sides 😂
Just follow the first trick, log second polynomial trigonometric trigonometric quadrangular
My math teacher was very simple about this, "try to find what would be the most convenient way to solve it"
3:15 5:50 10:20 12:32
I use the easy method one not like ilate or liate . We can get same answer by any method.
WHAT TO INTERGRATE USE *DETAIL* METHOD D-DX, E- EXPONENT,T -TRIG,A-ALGEBRA, INVERSE,LOG
😃😃😃😃WATCHING FROM SOUTH AFRICA
1:53 blackpenredpen becomes blackpenredpenbluepen 😂
One thing that I like about studying the later math courses in university is that the teachers for those courses don't care how I solve an integral - they are basically like "just solve this integral in some way", and only care about the answer. I am even allowed to use Wolfram-Alpha for example problems during classes if I feel like it, lol.
That's pretty much what you'd do in the real world anyway. Unless you are competing in MIT's integration bee, or discovering new mathematics, chances are, you'll have access to Wolfram Alpha in any situation outside an academic exercise, where you'd need to compute an integral.
The reason teachers who initially teach the method, want you to use the method they teach, is to make it easy to grade. Once it's no longer the main substance of the topic you're working with, no one will care what specific method you choose.
In computer science, you might need to know how to strategically pick the method that is least computationally intensive, to make your program more efficient, when it will have to do hundreds if not thousands of repeats of it, but that's the only time I can see a right or wrong choice of method, when faced with two options that get you the same solution.
OMG the integral at 9:05 I just didn't solve it with LIATE I solved it using partial fractions can't believe it 😋😋😋😋 wish u can see my solution it gave the final answer directly 😂😂
you are a GOAT..... the beard makes sense now
The hardest integral I solved with integration by parts was x*sin(x)*e^x - I did it with a double D-I-method, idk if it could have been easier
There are two ways for this triple product of x, sin(x), and e^x. One method is to use complex numbers, to rewrite sin(x)*e^x as a sum of two complex exponential functions, which will be i/2*e^([1 - i]*x) - i/2*e^([1 + i]*x). Pull the i/2 out in front as a constant, and assign constants N = 1 - i and P = 1 + i, to simplify our writing. Now you can integrate i/2*x*[e^(N*x) - e^(P*x)], with only one IBP table, after pulling i/2 out in front as a constant, and it is a simple ender. Then convert it back to the real world.
Another way you can do it, is to assign J and K such that:
J = integral sin(x)*e^x dx, and
K = integral cos(x)*e^x dx
Suppose we've previously solved these two integrals. The results are:
J = 1/2*[sin(x) - cos(x)]*e^x
K = 1/2*[sin(x) + cos(x)]*e^x
We'll use J & K, as we construct our IBP table. Let sin(x)*e^x be integrated, and x be differentiated:
S _ _ D _ _ I
+ _ _ x _ _ sin(x)*e^x
- _ _ 1 _ _ J
+ _ _ 0 _ _ integral J dx
Construct our result:
x*J - integral J dx
Since J is a linear combination of sin(x) and cos(x), this means integral J dx is a linear combination of J & K:
integral J dx =
1/2*J - 1/2*K
Thus, our result is:
x*J - [1/2*J - 1/2*K] =
(x - 1/2)*J + K/2
Fill in J & K:
(x - 1/2)*1/2*[sin(x) - cos(x)]*e^x + 1/4*[sin(x) + cos(x)]*e^x
Consolidate and simplify, and add +C:
1/2*[x*sin(x) - x*cos(x) + cos(x)]*e^x + C
I mean ILATE is easier to memorize and teach to students and u can identify the integral faster rather than trying to find which one can be integrated. Also integrals having both as same function are rarely seen in school level math at least in my schools
"wudanibinais"gets me everytime
ILATE is not useful at all
It's always better to understand the problem by intuition
This helps more
7:15 how did you know to stop there?
Was anyone else taught ILATE instead?
Yes
Someday I want to learn english totally to learn about Calculus 2. It's too difficult for me. TY for your videos
Remember when bprp had a whole university whiteboard to make his videos?
Well for the integral(ln(x)/(1+ln(x))^2 ,x) you can solve this with just u sub and integration by parts using the LIATE method. No need for creativity on that one
Let u = 1+ln(x)
integral( ln(e^(u-1))*e^(u-1)/u^2 ,u)
this simplifies and expands to
integral(e^(u-1)/u ,u)-integral(e^(u-1)/u^2 ,u)
then integral(e^(u-1)/u^2 ,u) use integration by parts by integrating 1/u^2 turning into -e^(u-1)/u+integral(e^(u-1)/u ,u)
this means the integral(e^(u-1)/u ,u)-integral(e^(u-1)/u^2 ,u)=integral(e^(u-1)/u ,u)-( -e^(u-1)/u+integral(e^(u-1)/u ,u))
that simplifies to e^(u-1)/u
putting x back you get the x/(1+ln(x))
Nah it's wrong. You get ((u-1)×e^(u-1))/u² which again needs to be solved with a "creative" integration by parts😂
@@chanuldandeniya9120 Expand the expression (u-1)*e^(u-1)/u^2 into two fractions and you get e^(u-1)/u-e^(u-1)/u^2. One of these fractions can be integrated separately as I have shown above using Integration by parts. I should have made the step where I separated them more clear but yes what you have is an intermediate step. I guess what makes my solution creative is that the integrals cancel out rather than being fully computed.
There is somerhing called the reduction formula in integration so reduction formulae is a special case for integration by parts
For most of our IBP problems, when we have a transcendental function and a power function, if the derivative of the transcendental function becomes algebraic, then it is u, if it stays transcendental, then it is with dv.
You can with integral zero-infinity x^2 e^-x^2 /1-e^-x^2 ?
Sir you are the lord of mathematics 😀😀🥳🥳
Are you in iitb ?
At 10:00 assume lnx to be t. dx equal to e^t dt and then apply by parts
THAT'S IT
you better remove your beards.
Integrals will always remain my favorite part of Calc I and II
They’re so fun to do! ❤️🥳
🧐😈Canon
I find derivatives more fun imo
@@kepler4192 ok. differentiate sin(cos(tan(csc(sec(cot(sinh(cosh(tanh(csch(sech(coth(arcsin(arccos(arctan(arccsc(arcsec(arccot(arcsinh(arccosh(arctanh(arccsch(arcsech(arccoth(x))))))))))))))))))))))))
in respect to x
Right LIATE rule is (3rd example) assume secx a and assume sec^2x b and solve [a (integral b)-[integral (derivative a).(integral b)]]
K genieass