without evaluating the integral, is the derivative of the function $\displaystyle \int_{2}^{x} e^{t^{2}-2} dt$ just $\displaystyle e^{t^{2}-2}$?

Thanks in advance for any assistance recieved :)

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- Nov 10th 2008, 09:30 PMtsal15simple derivative but confusing me
without evaluating the integral, is the derivative of the function $\displaystyle \int_{2}^{x} e^{t^{2}-2} dt$ just $\displaystyle e^{t^{2}-2}$?

Thanks in advance for any assistance recieved :) - Nov 10th 2008, 09:37 PMajj86Response
- Nov 10th 2008, 09:47 PMtsal15
Hey ajj86,

First I gotta say, you've been very helpful within the period of 20mins, more than some others on this forum have been in 20days.

MMM, I believe that you are implying the fundamental theorm of calculus whilst using antidifferentiation techniques aka finding the area under graph aka evaluating the integral...? I'm aware it would be a more perfect answer (can u get more perfect than perfect?), but the question strictly states that i must: "Without evaluating the integral, find F'(x) where F(x) = $\displaystyle \int_{2}^{x} e^{t^2-2} dt$" so having said that would $\displaystyle e^{t^2-2}$ be the answer?

Thank you

tsal15 - Nov 10th 2008, 09:52 PMajj86Response
I just found this:

But, I'm not really sure as to whether the answer I gave is correct. Let me do a little more investigating. - Nov 10th 2008, 10:01 PMtsal15
- Nov 10th 2008, 10:03 PMajj86Response
According to the form given, I'm thinking the answer would be:

e^(x^2 - 2) - Nov 10th 2008, 10:06 PMtsal15
- Nov 11th 2008, 06:05 AMajj86Response
Tsal15, I apologize for not replying last night. I'm not trying to take the easy way out, but I think this might give a better explanation than I can give:

Calculus Facts: Derivative of an Integral

Let me know what you think. - Nov 11th 2008, 06:25 AMMathstud28
This is just a simple application of the Second Fundamental Theorem of Calculus

$\displaystyle \frac{d}{dx}\int_a^{g(x)}f(t)dt=f(g(x))\cdot{g'(x) }$

The proof of this is most likely beyond the scope of your class. - Nov 12th 2008, 04:04 AMtsal15
- Nov 12th 2008, 04:19 AMtsal15
- Nov 12th 2008, 01:18 PMMathstud28
Wikipedia actually has a pretty non-analysis friendly version. Look here

__Fundamental theorem of calculus - Wikipedia, the free encyclopedia__