# Integration of an exponential function

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How to Integrate it:::

∫e^(ax²+bx+c)dx

Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.

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I am posting for first time in HW section If I did any mistake please please don't remove my post but edit it the way you want Mod Uncle. Thank youuu :D

Khashishi
That can't be solved in general with elementary functions. It requires the error function. https://en.wikipedia.org/wiki/Error_function
The imaginary error function is
$\int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \mathrm{erfi}(x)$

If you complete the square
$ax^2+bx+c=a(x+d)^2+f$
$d=\frac{b}{2a}$
$f=c-\frac{b^2}{4a}$
Then you get
$\int e^{a(x+d)^2+f} dx=\int e^{a(x+d)^2} e^f dx$
Then you can do some u substitution to put it into the form of the error function

That can't be solved in general with elementary functions. It requires the error function. https://en.wikipedia.org/wiki/Error_function
The imaginary error function is
$\int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \mathrm{erfi}(x)$

If you complete the square
$ax^2+bx+c=a(x+d)^2+f$
$d=\frac{b}{2a}$
$f=c-\frac{b^2}{4a}$
Then you get
$\int e^{a(x+d)^2+f} dx=\int e^{a(x+d)^2} e^f dx$
Then you can do some u substitution to put it into the form of the error function
Umm.... Looks like probably I haven't learn this function, by the way I'm in 12th year of High School..

Ray Vickson
Homework Helper
Dearly Missed
How to Integrate it:::

∫e^(ax²+bx+c)dx

Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.
To add to what Khashishi said in post #3, it is PROVABLE that no elementary antiderivative exists for $a \neq 0$ in your integral; that is, no finite expression can possibly exist for the antiderivative in the standard functions--powers, roots, trig functions, exponentials and the inverses of all these. Of course, there are non-finite expressions---such as infinite series and the like---that give the antiderivative, but no finite formula. Even if you allow yourself to write a formula 10 million pages in length, you still could not do it!

Let me emphasize: that result is a rigorously-proven fact. No matter how smart you are or how long you search, you can never find what you are looking for.

Google "non-elementary integration"; for example, see
http://www.sosmath.com/calculus/integration/fant/fant.html
for the basic facts and
https://www.math.dartmouth.edu/~dana/bookspapers/elementary.pdf
for some proofs.

Last edited:
Mark44
Mentor
Thread moved. Please post questions involving integration or differentiation in the Calculus & Beyond section, not the Precalc section.