Are you using the same data from the integration tutorial or using your own data? How many cells/genes are in your dataset?
Seurat's default integration method (CCA) is known to be runtime/memory intensive. It can handle large datasets but may require lots of CPU cores/memory. I have found that the amount of RAM also seems to increase as more CPU cores are used. That said, I have been able to run the integration tutorial with the included dataset on an 8GB laptop.
The easiest solution is to simply use more resources when integrating: increase the amount of RAM and consider using multiple cores (there is a separate Seurat tutorial on parallelization). If you are encountering runtime/memory issues and you cannot scale up your resources, I would suggest looking at alternative integration methods:
- Seurat includes another integration approach, RPCA which is supposed to be less memory intensive at the cost of being more conservative with integration
- There are several other integration methods that provide similar results. I have used Harmony and found that it requires significantly less resources than the Seurat integration approaches (both CCA and RPCA). A quick google search of single cell RNA-seq integration methods will turn up other popular methods with benchmarks to compare and contrasts different approaches.