how are these two integrals related??
Вставка
- Опубліковано 11 чер 2024
- 🌟Support the channel🌟
Patreon: / michaelpennmath
Channel Membership: / @michaelpennmath
Merch: teespring.com/stores/michael-...
My amazon shop: www.amazon.com/shop/michaelpenn
🟢 Discord: / discord
🌟my other channels🌟
mathmajor: / @mathmajor
pennpav podcast: / @thepennpavpodcast7878
🌟My Links🌟
Personal Website: www.michael-penn.net
Instagram: / melp2718
Twitter: / michaelpennmath
Randolph College Math: www.randolphcollege.edu/mathem...
Research Gate profile: www.researchgate.net/profile/...
Google Scholar profile: scholar.google.com/citations?...
🌟How I make Thumbnails🌟
Canva: partner.canva.com/c/3036853/6...
Color Pallet: coolors.co/?ref=61d217df7d705...
🌟Suggest a problem🌟
forms.gle/ea7Pw7HcKePGB4my5
6:47 This should be arctan(exp(x)) + arctan(exp(1 + (-1) - x)) instead, which is pi/2 and not 0.
Yes. I agree.
Since arctan function is an odd function, he used the principle that f(-x) = -f(x). Hence he moved the negative before the second arctan expression and then the arctan (e^x) cancel.
I was wondering about that. Thank you.
Trying to evaluate this, and can't figure it out. I can show that it is constant by differentiating and showing that the derivative is zero, but evaluating and getting pi/2 is not working. Could you point me in the direction of some help?
Okay, nevermind, wow that was simple.
this is the best integral trick that i've seen on the channel in over a year
Actually the integral of 1/(9+3^x) is elementary.
It can be easily taken by substitution t=9+3^x, resulting in 1/ln(3) ∫dt/(t(t-9)) .
In a more general case: ∫dx/(a^n+a^x) = - ln(a^(n-x)+1) / a^n ln(a) or - log_a (a^(n-x)+1) / a^n.
Integrating in the interval (0, 2n) gives n/a^n.
As about arc tan(e^x), the integral is non-elementary, so the trick really helps.
In more general case f(x) = arc tan(e^g(x)) where g(x) is an odd function. For a symmetric interval, like (-a, a) the value f(x)+f(-x) is always pi/2, no matter what g(x) is, so the value of the integral is always pi*a/2.
This was very, very cool. So cool it was backflip worthy.
Thank you, professor.
the value is equal to the length of the interval times the average of the function values at the endpoints. but because of the symmetry, that is equal to the length of the interval times the value of the function at the midpoint
Would be nice to see graphically what the family of curves that satisfy the condition look like.
Since you have to compute the constant, the best way to finish the integral off is probably to compute it as (b-a)/2*constant.
A variant of the "king property" of integration.
What is the king property?
@@zlodevil426 int(f(x)dx) from a to b = int(f(a+b-x)dx) from a to b
@@zlodevil426 according to my research, the properties known as king's rule, queen's rule and jack's rule are actually known by those names only in India
Pretty much
@@zlodevil426 integral from a to b of f(x) dx is equal to integral from a to b f(a+b-x) dx
Learned this from my cal 3 class a year ago but your explanation cemented my understanding on this
This feels like a version of the intermediate value theorem.
Not really, it uses a symmetry property of definite integrals (sometimes called the king's property): (a to b)∫f(x)dx = (a to b)∫f(a+b-x)dx; and then considers cases where f(x)+f(a+b-x) nicely reduces to a constant so we can get a nice formula for these cases
The last integral is easy if you factor 1/9 out make the substitution u = x-2 and split the integral up.
Yea the second example has an elementary antiderivative, so it can be solved by just using fundamental theorem of calculus without needing to invoke anything too fancy but I guess he wanted to display this theorem
Why is the exponential function inside of the arctan not part of the total function here?
Feels like you'd first need a substitution, t =exp(x), but that seems like it's ruin the condition too.
The zero value would imply that the integral evaluates to zero. Also, for anyone who doesn't get why the value is pi/2, it is because, more generally, arctan(t) + arctan(1/t) = pi/2, which is as a consequence of cotangent being the inverse of tangent. Here, t = e^x
@@minamagdy4126 Ah right. He wrote it down wrong.
arctan(exp(x)) + arctan(exp(-1+1-x)) = arctan(exp(x)) + arctan(1/exp(x)) = pi/2
Isn't it just integrating an odd function but with its center shifted to (a+b)/2 ?
Mostly. You have to shift the x-coordinate of the center from 0 to (a+b)/2 and the y-coordinate from 0 up (or down) to whatever the constant value of the sum of f(x)+f(a+b-x). Since both those effects are easy to do with the integral (the first being a linear substitution and the second just adding a constant to the integrand), this gives a pretty easy way to prove the statement from the well known statement int_(-c)^c g(x) dx = 0 for g odd.
More fun with functions?
examples on ℝ
f(x) = x for x irrational OR f(x) = -x for x rational
Or even ... f(x)
= x for x irrational
= -x for x rational with fraction part divisible by 2 (equivalently fraction part congruent to zero modulo 2)
= -x^3 for x rational with fraction part congruent to one modulo 2 ; condition *&
= x^3 for x rational with fraction part divisible by 3
= -x^4 for x rational with fraction part divisible by 5
= x^2 for x rational fraction part not divisible by 2, 3 or 5 and excluding condition *& above
The condition that f is continuous doesn't seem to be used in the proof of the theorem, even implicitly. Is there some reason it's needed?
Sir please provide lectures on sieve theory
Hmmm okay - I think this function will not satisfy the integration type qualities but it seems interesting any way - so I though I would share.
Based on y = x or rather y(x)=x on the reals.
Turn this into two cases f(x) = x- 0.001 if x is rational and f(x) = x + 0.001 if x is irrational.
I may be mistaken and apologies if so but for some epsilon/delta f might appear to be continuous and for others f is totally not-continuous.
It seems an interesting function. A further complexity is
f(x(t)) = x(t) - ε for x(t) rational and f(x(t)) = x(t) + ε where ε:= ε(δ) where δ is taken to analytically test for differentiability or continuity or both
Say ε(δ)= δ/2 or a scaled value to confound tests for continuity or differentiablity.
What was the point of the "t" variable? I don't feel like I've been convinced that t can be equal to x and also equal to a+b-x, although I trust MP that it's fine.
For definite integrals, the letter doesn't matter. It has bounds, so the variable of integration will disappear, so it doesn't matter how you write them. It was because he wanted to substitute, so he made his substitution x to make it clear he was following the pattern, so he gave the second bit a different name to then turn it into x.
The formula :
∫ g(x) dx x=-a to a = (1/2) ∫ ( g(x)+g(-x) ) dx x=-a to x=a
That's a pretty contrived condition at the start.
The condition seems to be based off of the "King property" of a class of definite integrals.
This video is the first I've ever heard this called the "King property". I more see it as a shifting of odd functions. If you were to graph a function with the given condition and then translate its center ((a+b)/2,f((a+b)/2) to be at the origin, what you get is an odd function, and this integral result is just a shifting of the well known statement int_(-c)^c g(x) dx = 0 for g odd.
Your vlog has been such a source of comfort and inspiration for me, especially during this past year. I appreciate how honest and vulnerable you are in sharing your experiences, both the highs and the lows. Watching your vlog feels like catching up with an old friend, and I always come away feeling more connected to the world around me. Thank you for being a bright spot in my day. 🌞
Shut up bot