  We obtain expressions for second kind integrals on non-hyperelliptic
$(n,s)$-curves. Such a curve possesses a Weierstrass point at infinity which is
a branch point where all sheets of the curve come together. The infinity serves
as the basepoint for Abel's map, and the basepoint in the definition of the
second kind integrals. We define second kind differentials as having a pole at
the infinity, therefore the second kind integrals need to be regularized. We
propose the regularization consistent with the structure of the field of
Abelian functions on Jacobian of the curve. In this connection we introduce the
notion of regularization constant, a uniquely defined free term in the
expansion of the second kind integral over a local parameter in the vicinity of
the infinity. This is a vector with components depending on parameters of the
curve, the number of components is equal to genus of the curve. Presence of the
term guarantees consistency of all relations between Abelian functions
constructed with the help of the second kind integrals. We propose two methods
of calculating the regularization constant, and obtain these constants for
$(3,4)$, $(3,5)$, $(3,7)$, and $(4,5)$-curves. By the example of $(3,4)$-curve,
we extend the proposed regularization to the case of second kind integrals with
the pole at an arbitrary fixed point. Finally, we propose a scheme of obtaining
addition formulas, where the second kind integrals, including the proper
regularization constants, are used.
