Gas expansion through adiabatic turbine
You are aware that, from the open system version of the first law of thermodynamics, the shaft work per unit mass must be equal to the change in enthalpy per unit mass, correct?  So you need to determine the change in enthalpy per unit mass between the gas entering the turbine and the gas exiting the turbine.  Do we agree so far?

We know that enthalpy is a function of state.  So we need to figure out a convenient path between state 1 at 100 bar, 600 K, and state 2 at 20 bar ,445 K such that the equation provided can easily be applied to determine the change in enthalpy.  Consider the following 3 part path:  


Hold the temperature at 600 K and lower the pressure to say 1 bar, where the gas behaves as an ideal gas.  Use the equation provided with dT set to 0 to evaluate $\Delta h$ for this step
Hold the pressure constant at 1 atm, and use the ideal gas heat capacity to calculate $\Delta h$ for cooling the gas down from 600K to 445K.
Hold the temperature constant at 445K and raise the pressure from 1 bar to 20 bars.  Use the equation provided with dT set to 0 to evaluate $\Delta h$  for this step.


The sum of the enthalpy changes for these three steps adds up to the enthalpy change for the overall change.