Integrals of Braided Hopf Algebras

The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d^{\mathrm{\prime }}}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d^{\mathrm{\prime }}}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^d$-submodule $H^{drat}$ of $H^d$ is found. That is, $H^{drat}\cong {\int }_{H^d}^l\otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category ${(}_B^B\mathcal{Y}\mathcal{D},C)$ are given.