Large language models (LLMs) show promise for automated test generation, but most approaches target unit tests with mocked dependencies. Low-level integration testing instead exercises a class with its real, in-project dependencies, exposing faults involving object construction, API call sequences, and component interactions. Generating such tests is challenging because LLMs may lack project-specific knowledge (not knowing) or violate provided constraints (not following). We present IntTestGen, an LLM-based approach that combines context-enriched generation with constraint-enforced fixing. It mines dependency usage patterns from project code to guide test generation, then applies symbol-, protocol-, and iteration-level constraints during repair using a ClassIndex, a Markov typestate model, and experience memory. We evaluate IntTestGen against the state-of-the-art LLM-based baseline PANTA and search-based baseline EvoSuite on Defects4J and Deps4J, a new post-cutoff benchmark of recent Java repositories. Across the two benchmarks, IntTestGen improves line coverage by 19.99 and 22.69 percentage points, branch coverage by 24.90 and 15.78 points, and mutation score by 13.67 and 0.17 points, respectively. It also covers 378 and 55 additional lines of dependency code. Ablation results confirm that all major components contribute to performance.

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