$$\int_{-\infty}^{\infty} \frac{1}{2\pi} \exp\{ -\frac{1}{2} ((y-x)^2 + x^2) \} dx$$

What I thought of doing was expand the inside

$$\int_{-\infty}^{\infty} \frac{1}{2\pi} \exp\{ -\frac{1}{2}y^2 + xy -x^2 \} dx$$

then I can take out the $ -\frac{1}{2}y^2$ part ... but how do I proceed?

I got line 2 of the image below, but how did they goto line 3?

enter image description here

  • 2
    $\begingroup$ square completion $\endgroup$ Oct 20 '13 at 6:06
  • $\begingroup$ He has gone away from this site after posting question . $\endgroup$ Oct 21 '13 at 5:43

Hope the following answer helps .

enter image description here


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