It has become increasingly clear that recommender systems overly focusing on short-term engagement can inadvertently hurt long-term user experience. However, it is challenging to optimize long-term user experience directly as the desired signal is sparse, noisy and manifests over a long horizon. In this work, we show the benefits of incorporating higher-level user understanding, specifically user intents that can persist across multiple interactions or recommendation sessions, for whole-page recommendation toward optimizing long-term user experience. User intent has primarily been investigated within the context of search, but remains largely under-explored for recommender systems. To bridge this gap, we develop a probabilistic intent-based whole-page diversification framework in the final stage of a recommender system. Starting with a prior belief of user intents, the proposed diversification framework sequentially selects items at each position based on these beliefs, and subsequently updates posterior beliefs about the intents. It ensures that different user intents are represented in a page towards optimizing long-term user experience. We experiment with the intent diversification framework on one of the world's largest content recommendation platforms, serving billions of users daily. Our framework incorporates the user's exploration intent, capturing their propensity to explore new interests and content. Live experiments show that the proposed framework leads to an increase in user retention and overall user enjoyment, validating its effectiveness in facilitating long-term planning. In particular, it enables users to consistently discover and engage with diverse contents that align with their underlying intents over time, thereby leading to an improved long-term user experience.