Simple exponential rule question having to do with integration

[Asphyxiated]

Homework Statement

This problem is from a first order differential equation i have been playing with but the question does really have anything to do with the problem itself. I can solve it just fine, I just can't resolve the final integral. I will write what I am talking about in section 3.

Homework Equations

(a^{b})^{c}=a^{b*c}

The Attempt at a Solution

So let's say we have a function we want to integral like this:

\int e^{x^{2}} dx

From the identity above it seems like i could change that to:

\int e^{2x} dx

but using a math program or my calculator it can easily resolve the e^{2x} but the e^{x^2} contains erf functions and it is certainly not straight forward. So I guess the question is if I can rewrite the first integral as the second integral or is:

e^{x^{2}}

different than:

(e^{x})^{2}

If they are different I am not sure how you would ever know that you couldn't make this switch unless you were told that the parentheses were intentionally left out to indicate that you can not make this switch.

I mean i understand that without the parentheses the order of operations is basically 'top down' while the parentheses will cause it to be evaluated from the 'bottom up' but when would you make this distinction?

If my question is not clear just let me know and I will try to explain it again in a different way.

 

[SteamKing]

There is a small flaw in your reasoning: x^2 is not equal to 2*x

e^(x^2) is not equal to e^(2*x)

 

[SammyS]

The identity is (a^{b})^{c}=a^{b*c} it is not a^{(b^{c})}=a^{b*c}

\displaystyle e^{x^{2}}=e^{x\cdot x}=\left(e^x\right)^x\,.

SK beat me to it. !

 

[Asphyxiated]

ah, ok, that makes it much clearer, I see where i was misinterpreting what was actually being stated.

Thanks both you!

 

[HallsofIvy]

Note that a^{b^c} should always be interpreted as a^{(b^c)} because (a^b)^c would be more reasonably written as a^{bc}.