Let the original intensities at points $A$ and $B$ be $I_A$ and $I_B$.

Given: $r=\frac{I_A}{I_B}=2$

A set of two polaroids is placed just before point $B$.

The source emits unpolarized light.

Effect of first polaroid:

For unpolarized light, intensity after the first polaroid becomes $I_1=\frac{I_B}{2}$

Effect of second polaroid:

The angle between the pass-axes is $45^\circ$. By Malus' law, $I_2=I_1\cos^2 45^\circ$ $I_2=\frac{I_B}{2}\cdot \frac{1}{2}=\frac{I_B}{4}$

So the new intensity at $B$ is $I'_B=\frac{I_B}{4}$

New ratio:

Intensity at $A$ remains unchanged, so $r' = \frac{I_A}{I'_B} = \frac{I_A}{I_B/4} = 4\cdot \frac{I_A}{I_B}$

Since $\frac{I_A}{I_B}=2$, $r' = 4\times 2 = 8$

Therefore, the required integer answer is $\boxed{8}$