I'm new to studying z-scores and I've been told that for a gaussian statistic, around 95% of the values lie within the area two standard deviations above and below the mean, which (in accordance to my interpretation) would imply,

${\int}_{\mu -2\sigma}^{\mu +2\sigma}A{e}^{-((x-\mu )/\sigma {)}^{2}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x=0.95\ast {\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}A{e}^{-((x-\mu )/\sigma {)}^{2}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x$

Firstly, am I correct in my presumption? and secondly, is there any way to calculate the integral on the left to prove this point mathematically?

${\int}_{\mu -2\sigma}^{\mu +2\sigma}A{e}^{-((x-\mu )/\sigma {)}^{2}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x=0.95\ast {\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}A{e}^{-((x-\mu )/\sigma {)}^{2}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x$

Firstly, am I correct in my presumption? and secondly, is there any way to calculate the integral on the left to prove this point mathematically?