\(\displaystyle C = \int_{T_{low}}^{T_{high}} \frac{1}{\left(h' - h_a\right)} dT\)

where \(\displaystyle h'\) is a slowly varying, monotonically increasing function with temperature, T, and \(\displaystyle h_a\) is linearly increasing with a gradient \(\displaystyle m\). Both functions for \(\displaystyle h'\) and \(\displaystyle h_a\) can be evaluated for any known T and \(\displaystyle h' > h_a\) for all \(\displaystyle T_{low} \le T \le T_{high}\).

Now... according to a particular reference (I don't have a copy, but it's known as British Standard 4485... catchy!) the solution to this integral involves performing a Chebyshev approximation and then performing a rectangular integration on the pieces:

\(\displaystyle C = \frac{\Delta T}{4}\sum_{i=1}^{4} \frac{1}{\left(h'(T_i) - h_a(T_i)\right)}\)

\(\displaystyle \Delta T = T_{high} - T_{low}\)

where

\(\displaystyle T_1 = T_{high} - 0.1026728 \Delta T\)

\(\displaystyle T_2 = T_{high} - 0.4062038 \Delta T\)

\(\displaystyle T_3 = T_{low} + 0.4062038 \Delta T\)

\(\displaystyle T_4 = T_{low} + 0.1026728 \Delta T\)

I don't understand this approximation at all. Could someone please explain this to me (or point out some links or references that explain this outcome)?