# Thread: exponential integral

1. ## exponential integral

Dear all,
I have exponential integral with polynomial. I tried to solve it but I could not .
the integration is :

$\int_{-\infty}^{\infty} (ax^2+bx+c) e^{ax^2+bx+c} dx$

Can any one help me please.

2. Wolfram couldn't give me an answer...

Are you sure it's not

$\int_{-\infty}^{\infty}(2ax + b)e^{ax^2 + bx + c}\,dx$

because that can be solved with a substitution...

Actually on second thought, I believe that integral I mentioned is divergent... Hmmm...

3. hi,
thank you for the replay.
yes the integral that i wrote is correct. and if it is like what you wrote then you are writ that it can be solved with substitution.

any other suggestion please.

4. Originally Posted by abotaha
Dear all,
I have exponential integral with polynomial. I tried to solve it but I could not .
the integration is :

$\int_{-\infty}^{\infty} (ax^2+bx+c) e^{ax^2+bx+c} dx$

Can any one help me please.

Where has this integral come from?

5. ok, there are mistakes in the integral equation.
the correct one is :

$\int_{-\infty}^{\infty} (-ax^2+bx+c) e^{-dx^2+bx+f} dx$

sorry for this mis-writing.

6. Originally Posted by abotaha
ok, there are mistakes in the integral equation.
the correct one is :

$\int_{-\infty}^{\infty} (-ax^2+bx+c) e^{-dx^2+bx+f} dx$

sorry for this mis-writing.

You still haven't answered my question:

Originally Posted by Mr Fantastic
Where has this integral come from?

7. The integral comes from multiplication of two functions:

$f_1=\ln (\frac {1}{4\pi^2})- \frac {(y-\mu_y-\beta_{yx}(x-\mu_x) -\beta_{yz}( z-\mu_z)-\beta_{yw}(w-\mu_w))^2}{2 \sigma_y^2} - \frac {( x-\mu_x )^2}{2 \sigma_x^2} - \frac {(z-\mu_z)^2}{2 \sigma_z^2} - \frac {(w-\mu_w)^2}{2 \sigma_w^2}- \ln(\sigma_y) - \ln(\sigma_x) - \ln(\sigma_z)- \ln(\sigma_w)$
and
$f_2=\frac{1}{2 \sqrt{\pi} \sigma_y\sigma_m}\left(\sqrt{2}\sqrt{\sigma_y^2+\b eta{yx}^2\sigma_m}\\
e^T\right)$

where
$T=R+\left(\frac{(y-\mu_y+\beta_{yx}\mu_m-\beta_{yz}(z-\mu_z)-\beta_{yw}(w-\mu_w))^2}{2\sigma_y^2}+\frac{\mu_m^2}{2\sigma_m^2 }-
\frac{\frac{(y-\mu_y+\beta_{yx}\mu_m-\beta_{yz}(z-\mu_z)-\beta_{yw}(w-\mu_w))\beta_{yx}}{\sigma_y^2}+\frac{\mu_m}{2\sigm a_m^2}}
{2(\frac{\beta_{yx}^2\sigma_m^2+\sigma_y^2}{\sigma _y^2\sigma_m^2)}}\right)$

and
$R=\left(\frac {(y-\mu_y-\beta_{yx}(x-\mu_m) -\beta_{yz}( z-\mu_z)-\beta_{yw}(w-\mu_w))^2}{2 \sigma_y^2} - \frac {( x-\mu_m )^2}{2 \sigma_m^2}\right)$

then I simplified the multiplication result as it written above:
$\int_{-\infty}^{\infty}f_1f_2 dx$