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.

]]>I think you're thinking of e[sup]x²[/sup].

Anyway, every function can be integrated, it's just that not all of them can be integrated algebraically.

If you wanted to find ∫e[sup]x²[/sup]dx, 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.

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?

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