5. Let
$$
A=\{(m, n) \mid 0<m<2,0<n<2\} \text {. }
$$

Then for any $(m, n) \in A$, the equation in terms of $x$
$$
\frac{m}{4} x^{2}+x+n=0
$$

has real roots with a probability of ( ).
(A) $\frac{1+2 \ln 2}{4}$
(B) $\frac{1+\ln 2}{2}$
(C) $\frac{3-2 \ln 2}{4}$
(D) $\frac{1-\ln 2}{2}$
5. A.

Given that the equation $\frac{m}{4} x^{2}+x+n=0$ has real solutions, we know $\Delta=1-m n \geqslant 0 \Rightarrow n \leqslant \frac{1}{m}$.
Therefore, the required probability is
$$
P=\frac{1}{4}\left(1+\int_{\frac{1}{2}}^{2} \frac{1}{x} \mathrm{~d} x\right)=\frac{1+2 \ln 2}{4} .
$$