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#1 2006-01-23 18:22:06

mikau
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fundamental theorem of calculus part 2

Just read about this today. It stated that every continuous function has an antiderivative. My book then trailed off on something about integreting with a constant as the lower limit of integration. In the end I couldn't see there point, and they then simply said "we state the theorem without proof:"

I'm not sure what the point was. I reread it 4 times and couldn't quite see what conclusion it made. Now obviously, every continuous function encloses a certain area above and below the x axis, so some integral must exist.

But then it said some functions can't be integrated, like e^-2x dx. But the fundamental theorem of calculus garentees e^-2x dx has an antiderivative. Seems like a self defeating statement. If there is no function that can be differentiated to get  e^-2x dx, then how can it have an antiderivative?


A logarithm is just a misspelled algorithm.

#2 2006-01-24 02:00:54

mathsyperson
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Re: fundamental theorem of calculus part 2

Correct me if I'm wrong, but isn't it this?



I think you're thinking of e.

Anyway, every function can be integrated, it's just that not all of them can be integrated algebraically.
If you wanted to find ∫edx, you'd need to take each and every real value of x and work out the integral at that point using numerical methods. I'd advise telling a computer to do it.


Why did the vector cross the road?
It wanted to be normal.

#3 2006-01-24 02:27:31

Ricky
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Re: fundamental theorem of calculus part 2

But then it said some functions can't be integrated, like e^-2x dx. But the fundamental theorem of calculus garentees e^-2x dx has an antiderivative. Seems like a self defeating statement. If there is no function that can be differentiated to get  e^-2x dx, then how can it have an antiderivative?

If you mean e^x^2, then you're going to love this one.  There is an antiderivative of e^x^2.  But we can't integrate it.  There is a function of it's antiderivative, but it was a function that was unknown until we were investigating the antiderivative of e^x^2.  Here is an example you will understand:



That is the defintion of ln(x).  The same thing happens when you take the integral of e^x^2.  The function is defined by that integral.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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