= Solution
The nonsingular <bivariate normal distribution> has $\sigma_1,\sigma_2>0$ and $|\rho|<1$. Put $a=(x_1-\mu_1)/\sigma_1$ and $b=(x_2-\mu_2)/\sigma_2$. Completing the square gives
$$
a^2-2\rho ab+b^2=(b-\rho a)^2+(1-\rho^2)a^2.
$$
Consequently the joint <probability density function> factors as
$$
f_{X_1,X_2}(x_1,x_2)=
\frac{e^{-a^2/2}}{\sqrt{2\pi}\sigma_1}\,
\frac{\exp[-(b-\rho a)^2/(2(1-\rho^2))]}
{\sqrt{2\pi}\sigma_2\sqrt{1-\rho^2}}.
$$
For fixed $x_1$, the second factor is a normalized <normal distribution> density in $x_2$, with mean $\mu_2+\rho\sigma_2a$ and standard deviation $\sigma_2\sqrt{1-\rho^2}$. Its integral is one; explicitly substitute $w=(b-\rho a)/\sqrt{1-\rho^2}$ in the <Gaussian integral>. Thus \b[the marginal density is]
$$
\boxed{f_{X_1}(x_1)=\frac1{\sqrt{2\pi}\sigma_1}
\exp\left[-\frac{(x_1-\mu_1)^2}{2\sigma_1^2}\right].}
$$
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