The gap between the plates of a parallel plate capacitor of area A and distance between plates d, is filled with a dielectric whose relative permittivity varies linearly from $$\epsilon_1$$ at one plate to $$\epsilon_2$$ at the other. The capacitance of the capacitor is:

$$\epsilon(x) = \epsilon_1 + \left( \frac{\epsilon_2 - \epsilon_1}{d} \right)x$$

$$dC = \frac{\epsilon_0 \epsilon(x) A}{dx}$$

Since these slices are stacked between the plates, they are in series. For series combinations, we integrate the reciprocals ($$1/dC$$):

$$\frac{1}{C} = \int_0^d \frac{1}{dC} = \int_0^d \frac{dx}{\epsilon_0 \epsilon(x) A}$$

$$\frac{1}{C} = \frac{1}{\epsilon_0 A} \int_0^d \frac{dx}{\epsilon_1 + \left( \frac{\epsilon_2 - \epsilon_1}{d} \right)x}$$

$$\frac{1}{C} = \frac{1}{\epsilon_0 A} \cdot \left[ \frac{d}{\epsilon_2 - \epsilon_1} \ln\left( \epsilon_1 + \frac{\epsilon_2 - \epsilon_1}{d} x \right) \right]_0^d$$

$$\frac{1}{C} = \frac{d \ln(\epsilon_2 / \epsilon_1)}{\epsilon_0 A (\epsilon_2 - \epsilon_1)}$$

$$C = \frac{\epsilon_0 (\epsilon_2 - \epsilon_1) A}{d \ln(\epsilon_2 / \epsilon_1)}$$