jmc

Let the integer be $abcd$. We know that $$\begin{array}{rl}a+b+c+d& =14,\\ b+c& =9,\\ a-d& =1.\end{array}$$ Subtracting the second equation from the first, we get $a+d=5$. Adding this to the third equation, we get $$2a=6\Rightarrow a=3$$ Substituting this into the third equation, we get $d=2$.

Now, the fact that the integer is divisible by $11$ means that $a-b+c-d$ is divisible by $11$. Substituting in the values for $a$ and $d$, this means that $1-b+c$ is divisible by $11$. If this quantity was a positive or negative multiple of $11$, either $b$ or $c$ would need to be greater than $9$, so we must have $1-b+c=0$. With the second equation above, we now have $$\begin{array}{rl}c-b& =-1,\\ c+b& =9.\end{array}$$ Adding these equations, we get $2c=8$, or $c=4$. Substituting this back in, we get $b=5$. Thus the integer is $\boxed{3542}$.