Given relation between pressure and volume

The pressure varies with volume as $P = kV^3$ where $k$ is a constant.

Use the ideal gas equation

For $n$ moles of an ideal gas, $PV = nRT$

Substituting $P = kV^3$: $kV^3 \cdot V = nRT$ $kV^4 = nRT$

Hence, $T = \frac{kV^4}{nR}$

So, $V^4 \propto T$ and therefore $V = \left(\frac{nRT}{k}\right)^{1/4}$

Expression for work done

Work done is $W = \int_{V_1}^{V_2} P\,dV$

Using $P = kV^3$, $W = \int_{V_1}^{V_2} kV^3\,dV$ $W = k\left[\frac{V^4}{4}\right]_{V_1}^{V_2}$ $W = \frac{k}{4}(V_2^4 - V_1^4)$

Replace $kV^4$ using ideal gas law

Since $kV^4 = nRT$ we get $kV_1^4 = nRT_1, \qquad kV_2^4 = nRT_2$

Therefore, $W = \frac{1}{4}(nRT_2 - nRT_1)$ $W = \frac{nR}{4}(T_2 - T_1)$

Substitute temperatures

Convert to kelvin: $T_1 = 100^\circ C = 373\,K$ $T_2 = 300^\circ C = 573\,K$

So, $\Delta T = 573 - 373 = 200\,K$

Hence, $W = \frac{nR}{4}(200) = 50\,nR$

Final answer

The work done is $\boxed{50\,nR}$

So the required integer is $\boxed{50}$.