Sequential Recommendation (SRs) that capture users' dynamic intents by modeling user sequential behaviors can recommend closely accurate products to users. Previous work on SRs is mostly focused on optimizing the recommendation accuracy, often ignoring the recommendation diversity, even though it is an important criterion for evaluating the recommendation performance. Most existing methods for improving the diversity of recommendations are not ideally applicable for SRs because they assume that user intents are static and rely on post-processing the list of recommendations to promote diversity. We consider both recommendation accuracy and diversity for SRs by proposing an end-to-end neural model, called Intent-aware Diversified Sequential Recommendation (IDSR). Specifically, we introduce an Implicit Intent Mining module (IIM) into SRs to capture different user intents reflected in user behavior sequences. Then, we design an Intent-aware Diversity Promoting (IDP) loss to supervise the learning of the IIM module and force the model to take recommendation diversity into consideration during training. Extensive experiments on two benchmark datasets show that IDSR significantly outperforms state-of-the-art methods in terms of recommendation diversity while yielding comparable or superior recommendation accuracy.