Normally you'd be given the suggestion that a function must have an infinite limit of 0 for the infinite integral to have a chance to converge. cos(x²) and sin(x²) seem to shatter this.
We can assign I think we can assign a value for ∫cos(wt)dt and ∫sin(wt)dt by Laplace transform with s→0 ∫cos(wt)dt⇒s/(s^2+w^2)→0 ∫sin(wt)dt⇒w/(s^2+w^2)→1/w Ref dr peyam that released a video on Laplace transform exacly when you released that video
create a function f(t) such that f(t) is the integral from zero to infinity of cos(tx^2) , then take the laplace transform of that function and you get an integral you can solve. math 505 made a video on it
*DAY 2* : One calc equation : What's the irl use of finding the volume when we rotate f(x) about a line using the Disc method, the Washer method etc, when we can simply....... Take the object and submerge it just below a given level of water of volume *v1 cubed units* and see the new reading of reaching the new greater value of *v2 cubed units* , and get the volume of the object as *v2-v1 cubed units* ??? [Done by Archimedes' Principle]
1) not everything could be easily submerged (whether due to size or amount of water) 2) not everything we want to get the volume of exists in real life. It's faster to compute then to go out of our way to make a real model of it 3) you may want to only get the volume of a portion of it without cutting or otherwise modifying it, and it might not have uniform density
Unrelated to the video, but I’ve been trying to wrap my head around the function y = the xth root of x. How do I differentiate it, and how do I write it in terms of y. Also is it possible to anti-differentiate it. Mainly though I just want to figure out how to write that function in terms of y. I tried using inverse operations and got y^x = x. Then I took the log base y of both sides, but I came full circle and got x = log base y of x, which is no better than y^x = x. Furthermore I can find hardly any information about that specific function online.
You should study it as y=e^(ln(x)/x), and to get an inverse function, you should look into the Lambert's W function defined as being the inverse function of x.e^x it might help (not sure but it's really good for this kind of problem where there's an x in the base and the exponent)
Yes, I solved it by first doing t = 1/x, then differentiation under the integral (aka Feynman integration), with help from Euler's formula and the Gaussian integral
@@Samir-zb3xk But no i didnt do that , actually i did a U sub first U =1/x then i used feynman on the resulting integral . feynman gave us : F(a)=integral from 0 to infinity of sin(a* U^2)/U^2 wjen we derive we get a fresnel integral he soolved it in the video i used the result i integrated and i put a=1 ( hope you understood its hard to explain over a comment)
@@draaagoo7799 oh ok, yea thats pretty much the same thing I did. I just derived the result in the video using Euler's formula and the Gaussian integral
@@SparkDragon42 definitions are not debatable or sensitive to personal tastes: the fundamental difference between an integral (or a series) that diverges (that is, that has an infinite result) and one that does not exist is that the first remains divergent even if you generalize the definition of integral (or series), while with generalizations the integrals (or series) that have no result can become convergent: if, for example, you take the sum of Cesaro or Holder, all the divergent series according to the classical definition remain divergent, all those already convergent remain convergent, some of those without a result become convergent. Would you say that the integral of the Dirichlet function diverges? Or is it more correct to say that it does not exist according to Rienmann and instead it exists according to Lebesgue?
I used a double integral to solve a single improper integral
ruclips.net/video/QDLDMDYxQ-0/видео.html
Thank God, that Ramanujan spent at most 5 minutes on this problem so i dont have to spend 1 hour.
Ramanujan's master theorem and/or Mellin
@@bluu1939 when you look at his original proof of his master theorem, it actually makes intuitive sense. Still, I'd never have come up with it.
I didn't expect that I will encounter a calculus cliffhanger today
Nice pfp makes me feel watched
Normally you'd be given the suggestion that a function must have an infinite limit of 0 for the infinite integral to have a chance to converge. cos(x²) and sin(x²) seem to shatter this.
That's quite interesting because, for series, this does need to be true
Great Work✅
Lets go a series!
The last curve is just the heartbeat of someone dying...
great video, but why you still use Geogebra Classic 5 in 2024 lol
Bc it has sentimental values to me!
😃
It's a nice version! Often works better.
Which program is better?
What is your suggestion?
@@yusufdenli9363Desmos lol
We can assign I think we can assign a value for ∫cos(wt)dt and ∫sin(wt)dt by Laplace transform with s→0
∫cos(wt)dt⇒s/(s^2+w^2)→0
∫sin(wt)dt⇒w/(s^2+w^2)→1/w
Ref dr peyam that released a video on Laplace transform exacly when you released that video
I was about to write about Laplace transforms for the same reason. Videos released at the same time. This is a mathematical conspiration.
Thanks!
waiting on that next vid
did you just sponsor BMT (Berkeley Math Tournament)!?!!!
Yes, and I am excited to be back!
Will u be hosting integ bee like last yr
He is not pregnant but he never fails to deliver
Fresnel Integrals!
And explanation of how the value for (3) is known would be interesting.
You can use Euler's formula to convert sin/cos into exponential functions and if you know the Gaussian integral the rest is straightforward
create a function f(t) such that f(t) is the integral from zero to infinity of cos(tx^2) , then take the laplace transform of that function and you get an integral you can solve. math 505 made a video on it
I found the value of the integral of sin(1/x^2) to be sqrt(π/2)
the next video is about a combinatorics problem...
😂
answer is sqrt(pi/2)
*DAY 2* :
One calc equation : What's the irl use of finding the volume when we rotate f(x) about a line using the Disc method, the Washer method etc, when we can simply.......
Take the object and submerge it just below a given level of water of volume *v1 cubed units* and see the new reading of reaching the new greater value of *v2 cubed units* , and get the volume of the object as *v2-v1 cubed units* ??? [Done by Archimedes' Principle]
how will you know exactly how the volume changes depending on the radius?
@@unturneddWith the help of a graduated beaker, which will contain readings like 100ml then 125ml and so on and also between the degrees.
1) not everything could be easily submerged (whether due to size or amount of water)
2) not everything we want to get the volume of exists in real life. It's faster to compute then to go out of our way to make a real model of it
3) you may want to only get the volume of a portion of it without cutting or otherwise modifying it, and it might not have uniform density
Unrelated to the video, but I’ve been trying to wrap my head around the function y = the xth root of x. How do I differentiate it, and how do I write it in terms of y. Also is it possible to anti-differentiate it. Mainly though I just want to figure out how to write that function in terms of y. I tried using inverse operations and got y^x = x. Then I took the log base y of both sides, but I came full circle and got x = log base y of x, which is no better than y^x = x. Furthermore I can find hardly any information about that specific function online.
You should study it as y=e^(ln(x)/x), and to get an inverse function, you should look into the Lambert's W function defined as being the inverse function of x.e^x it might help (not sure but it's really good for this kind of problem where there's an x in the base and the exponent)
If infinity is integer and multiple of any number including 2π, then what could it mean?
But maybe not possible.
i think i solved it , is it sqrt of (pi over 2)
Yes, I solved it by first doing t = 1/x, then differentiation under the integral (aka Feynman integration), with help from Euler's formula and the Gaussian integral
@@Samir-zb3xk same but i used the fresnel integral insteqd of gausian
@@draaagoo7799 yea you can do the Fresnel integrals by making sin/cos into exponential functions through Euler's formula then Gaussian
@@Samir-zb3xk But no i didnt do that , actually i did a U sub first U =1/x then i used feynman on the resulting integral . feynman gave us : F(a)=integral from 0 to infinity of sin(a* U^2)/U^2 wjen we derive we get a fresnel integral he soolved it in the video i used the result i integrated and i put a=1 ( hope you understood its hard to explain over a comment)
@@draaagoo7799 oh ok, yea thats pretty much the same thing I did. I just derived the result in the video using Euler's formula and the Gaussian integral
The integrals 1 and 2 do not diverge, they do not exist. In the same sense that the infinite sum of alternating addends 1 and -1 does not exist.
That's what diverging means. To diverge is to not converge, and if it doesn't exist, then it doesn't converge, so it diverges.
@@SparkDragon42 no, "diverge" is not the same that "not converge", "diverge" means "go to infinity".
@@VideoFusco let's agree to disagree as our definitions are obviously different
"divergence by oscillation"
@@SparkDragon42 definitions are not debatable or sensitive to personal tastes: the fundamental difference between an integral (or a series) that diverges (that is, that has an infinite result) and one that does not exist is that the first remains divergent even if you generalize the definition of integral (or series), while with generalizations the integrals (or series) that have no result can become convergent: if, for example, you take the sum of Cesaro or Holder, all the divergent series according to the classical definition remain divergent, all those already convergent remain convergent, some of those without a result become convergent. Would you say that the integral of the Dirichlet function diverges? Or is it more correct to say that it does not exist according to Rienmann and instead it exists according to Lebesgue?
Where the video of this monster integral 😢🎉
Is sqrtpi /2??
Yes!
Video will be up later
@@blackpenredpen I use integrals in my channels too :)
W
we need to stop him when it's not to early
bro i am your biggest fan from india. please help your lil bro in solving triple submition ques