Question

int((n/(2*pi))^(1/2)*exp(-(1/2)*nx^2), x = -infinity .. infinity)

Answer 1

\[I=\large \int\limits_{- \infty}^{\infty} \sqrt{\frac{n}{2 \pi}} e^{\frac{-nx^2}{2}} dx\\then\\I^2=\large \int\limits_{- \infty}^{\infty} \sqrt{\frac{n}{2 \pi}} e^{\frac{-nx^2}{2}} dx \int\limits_{- \infty}^{\infty} \sqrt{\frac{n}{2 \pi}} e^{\frac{-ny^2}{2}} dy\\ =\large \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} \sqrt{\frac{n}{2 \pi}} e^{\frac{-n(x^2+y^2)}{2}} dx dy\]
change too polar coordinates
\[I^2=\large \int\limits_{0}^{2 \pi} \int\limits_{0}^{\infty} \sqrt{\frac{n}{2 \pi}} e^{\frac{-n(r^2)}{2}} rdr d \theta=\sqrt{\frac{\pi}{2n}}\\so\\ \large I=(\frac{\pi}{2n})^{1/4}\]