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In this note, we study the integral of the 1-form $\log x\frac{dy}{y}-\logy\frac{dx}{x}$ over certain plane curves defined by A-polynomials of knots. Itis quite surprising that a Chern-Simons type invariant of 3-manifolds, whichcan be geometrically computed, may be used to get the exact values of thoseintegrals. The arithmetic nature of these integrals is still unknown at themoment and deserved further investigation.

On the integral of $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over the A-polynomial curves