Finding fault in integral function definition: G(x) = int [1,x] e^x dx

[lokguy]

Hi!

I've been staring blind at this problem for quite a while now, but I can't figure it out. It is the following:

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What's wrong with the following definition of a function \(\displaystyle G(x)?\)

. . . . .\(\displaystyle \displaystyle G(x)\, =\, \int_1^x\, e^x\, dx\)

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I looked into improper integral but that got me nowhere. There must be some fault in the logic of the notation or something but I just can't see it. Any hints or help would be greatly appreciated.

Thanks!

 

[Deleted member 4993]

Hi!

I've been staring blind at this problem for quite a while now, but I can't figure it out. It is the following:

"Whats wrong with the following definition of a function G?"

\displaystyle G(x)\, =\, \int_1^x\, e^{x}\, dx

I looked into improper integral but that got me nowhere. There must be some fault in the logic of the notation or something but I just can't see it. Any hints or help would be greatly appreciated.

Thanks!

\(\displaystyle \displaystyle {G(x)\, =\, \int_1^x\, e^{x}\, dx}\)
What are your thoughts regarding the assignment?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for

 

[HallsofIvy]

I'm not sure I would call it a "fault" (perhaps your teacher would), since it is so commonly done, but I would not write the same letter, "x", for both the final variable and for the integration variable. Instead I would write
\(\displaystyle G(x)= \int_1^x e^t dt= e^x- e\).

 

[lokguy]

I'm not sure I would call it a "fault" (perhaps your teacher would), since it is so commonly done, but I would not write the same letter, "x", for both the final variable and for the integration variable. Instead I would write
\(\displaystyle G(x)= \int_1^x e^t dt= e^x- e\).

I considered this but I agree that it's not a "fault" as far as I know. But I guess if you define the function F as a function of x, the end variable, then maybe it's not proper to integrate the original function with regards to x. ​Maybe it is the answer I'm looking for. Thanks for the help!

 

[lokguy]

\(\displaystyle \displaystyle {G(x)\, =\, \int_1^x\, e^{x}\, dx}\)
What are your thoughts regarding the assignment?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for

Thank you, I did read the rules before posting. I gave the only work I've done, other than that I've had thoughts about the answer I have disregarded but nothing that qualifies as "work".

To be more specific, I have gone down the road of treating is as a wrongly defined improper integral. In other words, instead of "x" as the upper limit, it should be "infinity" or a limit should be applied as "x approaches infinity". This was a bit of a hail mary, not really understanding what the fault was but it has been confirmed to me that this is incorrect. So I'm stuck after that point.

Many thanks for any tips!

 

[Dr.Peterson]

"Whats wrong with the following definition of a function G?"

View attachment 9457

I looked into improper integral but that got me nowhere. There must be some fault in the logic of the notation or something but I just can't see it. Any hints or help would be greatly appreciated.

What role does the variable x play in the integral? That is a key question to ask yourself.

 

[lokguy]

What role does the variable x play in the integral? That is a key question to ask yourself.

Well...I suppose the x in F(x) denotes that the integral equation is a function of x. The x in the integral limit denotes that the upper limit is some variable value. The x in e^x is the variable value of the function. So that leads me to think that we shouldnt integrate on the same variable as our limit, thus the function should be something like e^y to make proper sense?

 

[HallsofIvy]

Well...I suppose the x in F(x) denotes that the integral equation is a function of x. The x in the integral limit denotes that the upper limit is some variable value. The x in e^x is the variable value of the function. So that leads me to think that we shouldnt integrate on the same variable as our limit, thus the function should be something like e^y to make proper sense?

I'm feeling a bit ignored! That's exactly what I said in post #3!

 

[Dr.Peterson]

Well...I suppose the x in F(x) denotes that the integral equation is a function of x. The x in the integral limit denotes that the upper limit is some variable value. The x in e^x is the variable value of the function. So that leads me to think that we shouldnt integrate on the same variable as our limit, thus the function should be something like e^y to make proper sense?

Yes, the problem is that x is being used to represent two different things: one that varies within the integral and then is "used up", and another that is constant with regard to the integral, used as the limit of integration. As Halls said, you can overlook that, trusting that all the x's inside the integral represent one thing, and all those in the limits of integration or beyond are another; but it's cleaner to do as he did and use something like t inside. In more complicated cases, it would be entirely reasonable to interpret an x inside the integral as the same number used outside the integral, like this: \(\displaystyle \int_1^x e^{xt} dt \).

 

[HallsofIvy]

Yes, the problem is that x is being used to represent two different things: one that varies within the integral and then is "used up", and another that is constant with regard to the integral, used as the limit of integration. As Halls said,

Thank you for not ignoring me!

you can overlook that, trusting that all the x's inside the integral represent one thing, and all those in the limits of integration or beyond are another; but it's cleaner to do as he did and use something like t inside. In more complicated cases, it would be entirely reasonable to interpret an x inside the integral as the same number used outside the integral, like this: \(\displaystyle \int_1^x e^{xt} dt \).

No, the integral would be \(\displaystyle \int_1^x e^{t} dt \) without any "x" in the integrand.

 

[Dr.Peterson]

Thank you for not ignoring me!


No, the integral would be \(\displaystyle \int_1^x e^{t} dt \) without any "x" in the integrand.

You missed my point. I said this was a different situation (a "more complicated case"), not the same integral. The point is that an x inside could be the same as one outside. Conceivably, his integral could have been intended to be \(\displaystyle \int_1^x e^{x} dt \), which would be equivalent to \(\displaystyle e^{x} \int_1^x dt \), but that's highly unlikely. But I wanted an example where both x and t are in the integrand, which would be more likely to actually happen.