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Resolving dependencies among dialogue history is one of the main obstacles in the research on conversational question answering (QA). The conversational question rewrites (QR) task has been shown to be effective to solve this problem by reformulating questions in a self-contained form. However, QR datasets are limited and existing methods tend to depend on the assumption of the existence of corresponding QR datasets for every CQA dataset. This paper proposes a reinforcement learning approach that integrates QR and CQA tasks without corresponding labeled QR datasets. We train a QR model based on the reward signal obtained from the CQA, and the experimental results show that our approach can bring improvement over the pipeline approaches.