Temperature Dependent Specific Heat of Nitrogen
The function looks like its easy to differentiate or integrate. Should I be doing something involving calculus?


Yes, exactly!  The equation you should be using is the differential form of the enthalpy equation you are already using.  

$$dH = m~C_p(T)~dT$$

$$\int_{H_1}^{H_2} dH = m \int_{T_1}^{T_2} C_p(T)~dT  $$

As you can verify, if heat capacity is independent of temperature, i.e. a constant, the equation reduces to

$$H_2 - H_1 = m~C_p~\left (T_2 - T_1 \right ) \implies \Delta H = m~C_p\Delta T$$

This is the equation you tried to use originally.  However, since we derived this equation assuming a heat capacity that is independent of temperature, and your question explicitly involves dependence on the temperature, we should not use the already integrated equation.  Instead you'll have to do the integration yourself by substitution of temperature-dependent equation.