Let $f(x)=\ln(x)$.
Let $L$ and $M$ be two lines tangent to $f$ at the points $(t,f(t))$ and $(t+1,f(t+1))$.
Let $g(x)$ be the function who's graph is all the points of intersection of $L$ and $M$ for every $t\in(0,\infty)$.
Find the area between $f$ and $g$ in terms of known constants. That is, find $$\int_0^\infty g(x)-f(x) dx.$$
The decimal approximation is $0.3803307...$
Update: I derived the answer to be $\frac12 (\ln\pi+\ln2-\gamma)-\frac14$
It’s been a long time since I’ve looked at this problem so I’m just posting the answer I got for the record. Let me know if you’d like me to provide some details.