the trigonometric substitution skip
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- เผยแพร่เมื่อ 12 ก.ย. 2024
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Considering that trig functions are entangled with the Pythagorean Theorem, it makes sense that you can often use a Pythagorean type equation for the substitution (e.g. u ² = 1 - x ² is literally just the formula for the lengths of the sides of a right triangle of lengths u, x, and hypotenuse 1, and it’s also an easy way to prove that 1 = sin ²x + cos ²x as well. Likewise u ² = 1 + x ² is a triangle with a base length 1, height x, and hypotenuse u, making u linked to the tangent of the corresponding interior angle.)
So really in a way you’re just skipping the trig notation and going directly to playing with the corresponding squares of the lengths of the right triangle they represent on a unit circle.
Shouldn’t the second one end up as -1/3((1-x^2)/(x^2))^3/2+C
Looks like he intended to write x^-2 instead of x^2, just directly subbing in the expression after the second equals sign where u^2 was defined. (It's equivalent, but I think he was avoiding the more complicated expression.)
@@glenm99 that makes sense.
No, if he really meant that , the signs would have been the opposite. The substitution led to 1 - x^-2 istead of x^-2 - 1 . He just forgot to divide by x^2. No big deal
@@giorgioripani8469 Oh, maybe. I thought he brought the - sign outside.
(+1) I wonder why he never reviews his videos before uploading
Since I've learned about hyperbolic functions, I much prefer to use the substitution x = sinh(t) for integrals containing sqrt(1+x²), instead of x = tan(theta). Usually leads to an integral containing only exponential functions, and you don't need to use lots of tricky trig identities.
I always understand these videos, but would never be able to come up with these substitutions on my own.
That's Just how Maths and science works, though.
How tests go
The second substitution is absolutely crazy and out of the blue to me (I’ve been doing intervals for a long time). But the first one is quite normal. We didn’t even need to do u^2=1-x^2 as that might seem it out of the blue. When we have an odd exponent of x on the top, that’s a sign to use u sub normally. I used u=1-x^2.
At 8:30, you omitted the x^2 in the denominator when you substituted in for the final form.
This video has an excellent observation professor! But, I guess you got a mistake on the last integral result; probably, it should be or (-1/3)[((1-(x^2))/(x^2))]^(3/2) +C or (-1/3)[((x^(-2))-1)]^(3/2)+C, shouldn’t it?
A lot of it is practice and trial and error. Michael obviously has years of working these problems so he can quickly recognize potential substitutions. However he oresents a single substitution, there may be several that would work. As well he's not working these problems on the fly. You can tell he's looking at notes so he's already tried several substitutions until he find one that works and then presents that.
It would be fun to see him produce a video on his process of how he picks problems and works them out (mostly showing all the stuff that didnt work - this is where the real Math happens 99 failures for 1 success but a lot of fun exploration along the way).
As for a substitution the big thing you are looking for is a continuous function whose range is the domain of what you are integrating. So your integral is on some set S you ideally want a continuous function from P->S then you can integrate on P. Obviously there are A LOT of subtleties i just hand waived over.
I agree. Add in some cleverness and the "error" portion of trial and error becomes less likely.
With the second integral you should state that x > 0. Otherwise you have a sign change when the 1/x is placed under the root.
none of the variations of either example plays very differently from if you just do the trig sub. the first sub for all intents and purposes IS the standard trig sub, the second one is effectively u=csc(theta); the second problem played by the book you factor the trig into csc and cot to prosecute the resulting trig integral.
i do not believe there is any advantage to be gained here.
it's just binomial differential if you think about it. good video, Michael!
for those looking for the original article, its doi is 10.4169/college.math.j.48.4.284
Yes! You've finally given voice to something that has troubled me for years here. I *could* call it "how many symbols do you really need"....
9:12
It is interesting to see how his new substitutions are in some way "mimicking?" the trig identities. So for the first one, u^2 = 1 + x^2 when x = tan(t) then u becomes sec(t) and in the second one u^2 = 1/x^2 - 1, u becomes cot(t).
Conversely, you can sometimes think of the trig substitutions just being another way to write the u-substitutions.
i would prefer using u=sqrt(1+x^2) in the first one. as you stated it’s essentially equivalent, but it would avoid the absolute value issue that you ran into with sqrt(u^2).
merci sire excellent exercice , est il possible faire avec substitution hyperbolique
There's one weird substitution trick that works on any integral: Let u = F(x) such that dF/dx = f(x).
1du = f(x)dx
so we have the integral over 1 du, which is clearly u + c
Substituting back, we get F(x) + c -- Easy! 😎
I wouldn't have done the first one with trigonometric anyway, I'd just have done u=1+x^2, which also turns out quite simple. But for the second one, u=1-x^2 does NOT turn out simple, so it is possible that I would have tried trigonometric for that one, but it turns into cos^2/sin^4, which is also very painful, so at that point I would have probably paused to look for a third way. I love this trick of pulling one of the x's inside of the square root that you are showing us here, I'll definitely keep that in mind for the future, but I am having trouble seeing how to generalize the kinds of situations where this trick applies? It feels to me that this was one very lucky example, if it was any other exponent there instead of 4, it wouldn't have turned out that nice. What am I missing?
the cos^2/sin^4 = cot^2*csc^2 which is pretty straightforward from there.
Ah, this is true, thank you. I guess I always think of those as substituting either sin or cos and don't consider the other trig functions. Still, I like the idea of learning a path that doesn't have to go through trig, so the question remains, about whether this was just a lucky example or if it will often work.
The answer of second one should be -(1/3)(1-x²)^(3/2)+C (0
This was standard in my Calc class.
I liked that second integral. It came out so cleanly.
I suppose a math type question to ask may be: why do these substitutions work? What framework lies behind it and what can the framework tell us?
In other words: it is good that it works but why can we rely on it? Hmmm?
If you are asking why the substitution rules work at all, it is because of the Chain Rule in differential calculus. Substitution is essentially the chain rule in reverse. (Kind of.)
@@Notthatkindofdr I was thinking of things like morphisms, isomorphisms, structure and shape of solution sets.
Michael's example seems good background when abstract notions of isomorphic groups comes around.
GREAT ! Thank you !
Can you demonstrate an integral where you do a really non-obvious substitution. Like the met effect of three or something.
Integral on LHS substitution u=sqrt(x^2+1)
Integral on RHS by parts
great tips. thanks
I’m a big fan of his videos, even though he often makes small calculation errors, like here again 😁
I find it useful to, for example, instead of doing sin(theta) = x, do sin^-1 x = theta instead. That way you get straight to dx = dtheta sqrt(1-x^2). Plus there's no extra x shenanigans. I'm all about that polynomial lifestyle 🤙
Are you teaching calc 2 this semester? A lot of your videos are focused around that!
I am gonna be honest. The difficulty of trigonometric substitution for me was that it was always the section that ended up weather purposesily or not ensuring the teacher focused more on root memorization based elements than anything else
As well as infirectly them shutting down things that enabled thise techniques such as notes
I see your work was amaze, can you compare it with trig subs? Kindly, thank you sir.
If the calculation was using trigonometry subs, will be the result bs same?
Clever !
any source in college math journal？
Greetings from 🇦🇹🙂!
I am gonna get my teacher annoyed by using this
but i like the trig sub :(
Very good, cool. Thanks 😊
Boring teacher. Does not engage students