I haven't found any examples of boundless integrals suitable for introducing a parameter as you would do with textbook integrals but maybe that is part of the definition - that it only works on definite integrals
I've been self-teaching calculus for a while now and ran into DUTIS from a random video online. I understand the concept and how to do it, but I have trouble knowing what parameter to place and where to put it in the original integral.
Does anyone have any tips, advice, or help to offer with this? Thank you for reading and for your time :)
The Leibniz integral rule basically asks us to turn ,say a constant in an expression to a variable(say b) and define the Integral function as the function of the constant turned variable and differentiate it partially with respect to this b .
(ie) Example : if we are to find integral of say. I = x^5/ln(x) from 1 to 7
Here we turn 5 as n and differentiate with respect to n and etc .
Why does this work ?
How did Leibniz find this?
Is there any geometric interpretation (or) intuition for this ?
Can someone give me an example where using Leibniz rule would be useful.
I'm referring to the formula on this page
https://en.wikipedia.org/wiki/Leibniz_integral_rule
specifically i'm interested in what the meaning of the integral of the del x term could possibly represent. Is there some sort of fluid dynamics interpretation?
Hello! I've been struggling with a single problem all day. This one.
My professor suggested me to derivate twice at x, as you can see I did. But then I have the derivative at x of f(t), which gives 0. The problem is that it'll leave me with
f"(x)=0 and f(x)=c1*x+c2
When I try evaluating the first statement, it doesn't work. Even the integral does not converge.
I'm sure that all I need is to see a similar example, a link to somewhere I can read about that kind of equations, or just a term so I can do my search. I've tried searching for integral equations, and leibniz integral rule equations, but I couldn't find anything relevant.
Any comment will be appreciated, thank you.
We are told to think of Leibniz notation as one symbol rather than a fraction of two quantities. Can someone ELI5 how the chain rule, integration by substitution and separation of variables can be justified more rigorously than "cancel the dx's"?
Also I read that mathematicians came up with a more rigorous way to define dx and dy as differentials, so dy/dx could be a fraction of them. How could they use the chain rule and so on before it had a proper justification?
Sorry, it's too much to understand or explain for an A level student curious about the notation and no background in differential geometry so maybe the question is bad I don't know.
I read that Leibniz started calculus by first finding the integral, and Newton started with the derivative. Are there any records that show how they each came up with or saw the connection between the two?
So I've been learning the method of differentiating under the integral sign and I have managed to answer some questions where the parameter is easy to put into the integrand e.g. for int x^2 -1/logx dx I would let I(t) = int x^t - 1/logx dx
What should I define as I(t) for this integral int 8/(1+x^2 )^2 dx and are there any hints for questions like this in general?
Thanks for any advice.
I need to differentiate the following integral with respect to x (not x(bar)):
[3/Gamma(2/3)] (Integral from x to infinity) [x(bar)*exp(-x(bar)^3 ) ] dx(bar).
Gamma refers to the gamma function, I believe gamma(2/3) will just be constant.
Dummy variable confuse me a bit, which may be why I'm having so much trouble, not sure if I can push the d/dx into the integral.
Any help would be awesome and very much appriciated :). Thanks for reading!
By that i mean pulling a differentiation in/out of an integral.
Bare in mind im a maths/physics undergrad, so no real/complex analysis been studied, just looking for a bit of intuition.
My current intuition is along the line of, if the integral and its bounds are independant of the differentiation, we can use the Leibniz rule. Eg integral of whole real line in dx, contains function c(x,t), so d/dt can come in/out of integral. Can anyone correct this?
Hi,
while trying to understand connections and the covariant derivative I came across [this](https://www.youtube.com/watch?v=nEaiZBbCVtI) lecture by Frederic Schuller, where he describes the topic in a fairly non-abstract way (compared to his other lecture series at least...). At one point he mentions, that the following two versions of the Leibniz rule for connections are equivalent:
- D(T (x) S) = (DT) (x) S + T (x) DS
- D(T(w, X)) = DT(w, X) + T(Dw, X) + T(w, DX)
(where D is \nabla_X and (x) is a tensor product)
and I haven't been able to figure out why. Any help would be much appreciated.
Recently Iβve heard a lot about various members of the rules committee saying that they didnβt feel that certain cards in the format are healthy, yet they have a history of banning cards that they specifically didnβt like in their own local metas. Any time that Iβve ever seen criticism at this, they always fall back on βrule zeroβ should be a big discussion point in your playgroup. My question then becomes, if any problematic cards, or rather an unproblematic card can just be banned or unbanned at your own discretion, what justification is there in an entity such as the committee existing?
Edit: Give rationale that doesnβt include βestablishing a baselineβ, because if Wotc is making product for the format, a separate entity enforcing rules is further not needed.
As it turns out Iβm no Feynman and am struggling to understand how and why feynmans trick of differentiating under the integral works and having even more difficulty applying it myself. I was wondering if anyone knows any online resources or videos that can help it click, because Iβm pretty stuck and annoyed Any help is very much appreciated
Hello, I am trying to understand the substitution rule for indefinite integrals, and have a reduced example that I find confusing.
Say we wish to integrate 1/x dx. Of course the answer is ln(x).
We may also rewrite it as (1/-x) * -dx. This suggests the substitution:
u = -x
du = -dx
Now the integral is 1/u du, whose solution is ln(u). Back-substitute, we arrive at ln(-x), which is different from ln(x).
What went wrong? I think it's that true integral of 1/x needs to be understood as ln(|x|): the absolute value wipes out negation.
Thanks for any thoughts. This is not a homework problem: I'm trying to recall high school calculus on my own.
The Leibniz integral rule basically asks us to turn ,say a constant in an expression to a variable(say b) and define the Integral function as the function of the constant turned variable and differentiate it partially with respect to this b .
(ie) Example : if we are to find integral of say. I = x^5/ln(x) from 1 to 7
Here we turn 5 as n and differentiate with respect to n and etc .
Why does this work ?
How did Leibniz find this?
Is there any geometric interpretation (or) intuition for this ?
The Leibniz integral rule basically asks us to turn ,say a constant in an expression to a variable(say b) and define the Integral function as the function of the constant turned variable and differentiate it partially with respect to this b .
(ie) Example : if we are to find integral of say. I = x^5/ln(x) from 1 to 7
Here we turn 5 as n and differentiate with respect to n and etc .
Why does this work ?
How did Leibniz find this?
Is there any geometric interpretation (or) intuition for this ?
