Otherwise, I quite agree. I don't write unit tests. A complete set of integration tests gives me about 80% confidence in the correctness of code. Unit tests might bump that up to 90%, but you'll never get to 100% from testing alone and the substantial effort necessary to get there just isn't worth it.
Testing follows the classic 80/20 rule. Integration tests only take 20% of the effort required for full unit test coverage, but can give you the bulk (80%) of the benefits. For all but the most sensitive of applications, it's probably not worth it to put in the additional 80% effort for a mere 20% gain.