Let $X_1,X_2$ be correlated Gaussian random variables, centered, such that $\text{cov}(X_1,X_2) = \epsilon$, for $0<\epsilon<1$.
Calculate $\mathbb E [\max(X_1,X_2)]$.
I don't really know how to perform the calculation to its end. I first calculated the density. Let's right $\Gamma$ the covariant matrix of $(X_1,X_2)^\top$. Thus, one may find $\det \Gamma = 1-\epsilon^2$ and the density $f(x) = \frac{1}{2\pi\sqrt{1-\epsilon^2}}\exp\left(-\frac{1-\epsilon^2}{2} (x_1^2+x_2^2 - 2\epsilon x_1x_2)\right)$
Then, I tried to compute: $$ \begin{align} \mathbb E [\max(X_1,X_2)] &= 2\int_{\mathbb R}\int_{x_1}^{+\infty} x_2f(x)dx_2dx_1 \\&= \frac{1}{\pi\sqrt{1-\epsilon^2}}\int_{\mathbb R}\int_{x_1}^{+\infty}x_2\exp\left(-\frac{1-\epsilon^2}{2} (x_1^2+x_2^2 - 2\epsilon x_1x_2)\right)dx_2dx_1 \end{align} $$
But I don't know how to deal with the cross term in the exponential.
I assume that $X_1$ and $X_2$ have joint distribution $\mathcal N\left(0,\begin{bmatrix} 1&\varepsilon\\\varepsilon&1 \end{bmatrix} \right)$.
For any $a$ and $b$, $\max(a,b)=\frac{a+b}{2}+\frac{|a-b|}{2}$, applying this to your scenario you get \begin{align*} \mathbb E[\max(X_1,X_2)] &= \mathbb E\left[\frac{X_1+X_2}{2}\right] + \mathbb E\left[\frac{|X_1-X_2|}{2}\right]\\ &= \frac{\mathbb E[|X_1-X_2|]}{2} \end{align*} But now $X_1-X_2\sim\mathcal N(0,2(1-\varepsilon))$, it is known that for such a gaussian random variable, the expectation of it's absolute value is $2\sqrt{\frac{1-\varepsilon}{\pi}}$ (see this link).
Hence $\mathbb E[\max(X_1,X_2)]=\sqrt{\frac{1-\varepsilon}{\pi}}$.