When vehicles move, particularly at high speeds, friction between the vehicle body and the air can cause the accumulation of static electric charge on the surface of the vehicle. For vehicles carrying inflammable fluids, the presence of static electricity poses a significant risk as it can lead to sparking, which may cause combustion or an explosion if the fluid is highly flammable.
To mitigate this risk, metallic chains are often attached to such vehicles, ensuring that they touch the ground. The metallic chains provide a conductive path for the excess static charge to flow safely to the ground, thereby preventing the buildup of static electricity and reducing the risk of sparking.


Two resistors $2\,\Omega$ and $3\,\Omega$ are connected in the gaps of a bridge as shown in the figure. The null point is obtained with the contact of jockey at some point on wire $XY$. When an unknown resistor is connected in parallel with $3\,\Omega$ resistor, the null point is shifted by $22.5\,\text{cm}$ towards $Y$. The resistance of unknown resistor is ___ $\Omega$. 
The value of \[ \int_0^{2} \sqrt{\frac{x(x^2+x+1)}{(x+1)(x^4+x^2+1)}} \, dx \] is