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Let X be a complex algebraic manifold of dimension n+1 embedded in asufficiently higher dimensional complex projective space, and Y a generichyperplane section of X. We describe the mixed Hodge structure on H^p(X-Y,C)and the Hodge filtration of the middle primitive cohomology group H^n(Y,C)_0 ofY in terms of rational integrals on X. (Key words: Primitive cohomology,Rational integral of the 2nd kind, Generalized Poincare residue map, Hodgefiltration, Mixed Hodge structure)

Rational Integrals of the second kind on a complex projective manifold and its primitive cohomology