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$\begin{aligned} I & = \int_0^\pi \cos^{14} \theta \ d \theta & \small \color{#3D99F6}{\text{Since the integrand is symmetrical at }\theta = \pi /2.} \\ & = 2 \int_0^{\pi/2} \cos^{14} \theta \ d \theta \\ & = 2 \int_0^{\pi/2} \sin^0 \theta \cos^{14} \theta \ d \theta \\ & = B \left( \frac 12, \frac {15}2 \right) \\ & = \frac {429 \pi}{2048} \end{aligned}$

$\implies A+B+C \equiv 0+429+2048 \equiv 2477 \equiv \boxed{77} \text{ (mod 100)}$