How do I go from ∫_0^x f(t) dt to int_g(x)^h(x) f(t) dt?

[s3a]

Homework Statement

In other words, I'm trying to use the "historical" version of that part of the Fundamental Theorem of Calculus to obtain the "powerful" version.

Homework Equations

Chain rule (I think).

The Attempt at a Solution

I can't find how to do this in my book, and I have a strong feeling that this involves the chain rule, but I'm not too sure how to get my vague idea into writing, and any help in doing so would be greatly appreciated!

 

[LCKurtz]

Homework Statement

In other words, I'm trying to use the "historical" version of that part of the Fundamental Theorem of Calculus to obtain the "powerful" version.

Homework Equations

Chain rule (I think).

The Attempt at a Solution

I can't find how to do this in my book, and I have a strong feeling that this involves the chain rule, but I'm not too sure how to get my vague idea into writing, and any help in doing so would be greatly appreciated!

It is best to put the equations in the body of your post, not in the title. If I understand you, you are asking how this equation$$
\frac d {dx}\int_a^x f(t)~dt = f(x)$$implies the more general Leibnitz rule. You are correct that it involves the chain rule. For example say we want to differentiate$$
F(x) = \int_a ^{h(x)} f(t)~dt$$Here, you can let $u = h(x)$ so $F(x) = \int_a^u f(t)~dt$. Then using the chain rule:$$
F'(x) = \frac{dF}{du}\cdot \frac{du}{dx}=f(u)\cdot \frac{du}{dx}=f(h(x))h'(x)$$The lower limit is treated similarly.

 

[s3a]

Edit: Sorry, I double-posted.

 

[s3a]

Thanks for the answer.

Your assumption as to what I am asking is correct, and I agree that I should have also added the algebraic expressions in the body of the opening post, so I will add it there there as soon as this message has been posted.

From what you said, I don't understand how to justify replacing x with h(x). I get that h(x) = x is valid (because h is a function of x and x is such a function), but I don't see how we can assume that h(x) can be a function of x that is not just x.

Could you please explain the reasoning as to why the first equation you gave implies that all other functions of x (that are not just x) can be considered on the upper limit of integration for the second equation you gave?

 

[LCKurtz]

Thanks for the answer.

Your assumption as to what I am asking is correct, and I agree that I should have also added the algebraic expressions in the body of the opening post, so I will add it there there as soon as this message has been posted.

From what you said, I don't understand how to justify replacing x with h(x). I get that h(x) = x is valid (because h is a function of x and x is such a function), but I don't see how we can assume that h(x) can be a function of x that is not just x.

Could you please explain the reasoning as to why the first equation you gave implies that all other functions of x (that are not just x) can be considered on the upper limit of integration for the second equation you gave?

I can hardly make sense of this response. I didn't "replace" $x$ with $h(x)$. There is no law against writing a definite integral with variables in the limits, nor differentiating such an expression. You asked how to derive the differentiation formula for such a problem and I showed you.

 

[Ray Vickson]

Thanks for the answer.

Your assumption as to what I am asking is correct, and I agree that I should have also added the algebraic expressions in the body of the opening post, so I will add it there there as soon as this message has been posted.

From what you said, I don't understand how to justify replacing x with h(x). I get that h(x) = x is valid (because h is a function of x and x is such a function), but I don't see how we can assume that h(x) can be a function of x that is not just x.

Could you please explain the reasoning as to why the first equation you gave implies that all other functions of x (that are not just x) can be considered on the upper limit of integration for the second equation you gave?

What is the problem? The Fundamental Theorem says that if $F$ is an antiderivative of $f$, then
$$\int_0^{\text{anything not containing}\,t} f(t) \, dt = F(\text{anything not containing}\,t) - F(0).$$

 

[s3a]

Ray Vickson, my problem is (or was?) that I don't see how LUKurtz's first equation (in his first post), is saying that the upper limit of integration is “anything not containing t”.

It seems that what you guys are saying is that what was done was simply to recognize that it is just a general concept of integrals that anything that does not involve the variable being integrated can be in the limits of integration, and that there was not any kind of algebraic manipulation on the upper limit of integration, x, of the first equation LCKurtz gave (in first post), such that all that mattered from the first equation LCKurtz gave (in his first post), was merely to establish a relationship between differential and integral calculus.

Is this correct?

 

[Ray Vickson]

Ray Vickson, my problem is (or was?) that I don't see how LUKurtz's first equation (in his first post), is saying that the upper limit of integration is “anything not containing t”.

It seems that what you guys are saying is that what was done was simply to recognize that it is just a general concept of integrals that anything that does not involve the variable being integrated can be in the limits of integration, and that there was not any kind of algebraic manipulation on the upper limit of integration, x, of the first equation LCKurtz gave (in first post), such that all that mattered from the first equation LCKurtz gave (in his first post), was merely to establish a relationship between differential and integral calculus.

Is this correct?

I cannot figure out what you are asking. Anyway, you have been given the answers to your original question.

 

[s3a]

Sorry for being difficult to understand.

I can't be 100% sure, but I think I understood what I should be doing.

Thank you both for the help.