Dear all,
I have exponential integral with polynomial. I tried to solve it but I could not .
the integration is :
$\displaystyle $\int_{-\infty}^{\infty} (ax^2+bx+c) e^{ax^2+bx+c} dx$$
Can any one help me please.
Thanks in advance,
Wolfram couldn't give me an answer...
Are you sure it's not
$\displaystyle \int_{-\infty}^{\infty}(2ax + b)e^{ax^2 + bx + c}\,dx$
because that can be solved with a substitution...
Actually on second thought, I believe that integral I mentioned is divergent... Hmmm...
The integral comes from multiplication of two functions:
$\displaystyle $f_1=\ln (\frac {1}{4\pi^2})- \frac {(y-\mu_y-\beta_{yx}(x-\mu_x) -\beta_{yz}( z-\mu_z)-\beta_{yw}(w-\mu_w))^2}{2 \sigma_y^2} - \frac {( x-\mu_x )^2}{2 \sigma_x^2} - \frac {(z-\mu_z)^2}{2 \sigma_z^2} - \frac {(w-\mu_w)^2}{2 \sigma_w^2}- \ln(\sigma_y) - \ln(\sigma_x) - \ln(\sigma_z)- \ln(\sigma_w)$$
and
$\displaystyle $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)$$
where
$\displaystyle $T=R+\left(\frac{(y-\mu_y+\beta_{yx}\mu_m-\beta_{yz}(z-\mu_z)-\beta_{yw}(w-\mu_w))^2}{2\sigma_y^2}+\frac{\mu_m^2}{2\sigma_m^2 }-
\frac{\frac{(y-\mu_y+\beta_{yx}\mu_m-\beta_{yz}(z-\mu_z)-\beta_{yw}(w-\mu_w))\beta_{yx}}{\sigma_y^2}+\frac{\mu_m}{2\sigm a_m^2}}
{2(\frac{\beta_{yx}^2\sigma_m^2+\sigma_y^2}{\sigma _y^2\sigma_m^2)}}\right)$$
and
$\displaystyle $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)$$
then I simplified the multiplication result as it written above:
$\displaystyle $\int_{-\infty}^{\infty}f_1f_2 dx$$