1. For two coherent beams with intensities $I_1$ and $I_2$, the maximum and minimum intensities in interference are:

$I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2$ $I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2$

2. Here,

$I_1=4I, \qquad I_2=9I$

So,

$\sqrt{I_1}=\sqrt{4I}=2\sqrt{I}, \qquad \sqrt{I_2}=\sqrt{9I}=3\sqrt{I}$

3. Maximum intensity:

$I_{\max}=(2\sqrt{I}+3\sqrt{I})^2=(5\sqrt{I})^2=25I$

4. Minimum intensity:

$I_{\min}=(2\sqrt{I}-3\sqrt{I})^2=(-\sqrt{I})^2=I$

5. Difference between maximum and minimum intensities:

$I_{\max}-I_{\min}=25I-I=24I$

Given this difference is $xI$, we get

$xI=24I \Rightarrow x=24$

Therefore, the required integer is:

$\boxed{24}$

6. Comparison with stored correct answer:

Stored correct answer = 24

Our derived answer also = 24, so they agree.