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Well the title says it all:

Can someone please help me integrate:

$$\int_0^{500} e^{\frac{-(x-a)^2}{b^2}} \, dx$$

I need to understand how $e$ can be integrated when it is to the power of a polynomial. Another example that would help with my understanding is how someone would integrate:

$$\int_0^{500} e^{\frac{-x^2 + 4ax - 2a^2}{b^2}} \, dx$$

$a$ and $b$ are constants in this case.

Any help would be appreciated.

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    Does your integral have limits? As it is, it cannot be expressed in terms of elementary functions, but it can be expressed in terms of the [error function](http://en.wikipedia.org/wiki/Error_function) by making an appropriate substitution. However, if you include (appropriate) limits then you can get a closed-form expression.2012-09-14
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    Ooops forgot those. . Edited2012-09-14
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    You meant $-(x-a)^2$ in the exponential term. I guess it follows your last [question](http://math.stackexchange.com/questions/195639/transforming-a-continuous-function). Just use the expression with $\phi$ and compute $\phi$ using approximations (like Abramowitz & Stegun, see [this Wikipedia article](http://en.wikipedia.org/wiki/Normal_distribution#Generating_values_from_normal_distribution)).2012-09-14
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    Thanks again vanna. I thought maybe there was a way to calculate it, but if not then I will look more carefully into using a cdf.2012-09-14

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