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To fully utilize graphene's remarkable optical properties for optoelectronic applications, it needs to be integrated in planar photonic systems. Here, we demonstrate integration of graphene on silicon photonic circuits and precise measurement of the optical absorption coefficient in a graphene/waveguide hybrid structure. A method based on Mach-Zehnder interferometry is employed to achieve high measurement precision and consistency, yielding a maximal value of absorption coefficient of 0.2 dB/μm when graphene is located directly on top of the waveguide. The averaged results obtained from a large number of samples agree with theoretical model utilizing the universal ac conductivity in graphene. Our work provides an important guide for the design and optimization of integrated graphene optoelectronic devices.