Integral gutters go by different names. Yankee gutters is probably the best known of the terms, although in my area I also hear them called Dutch gutters.
The photo below gives a good example of what they look like.
Unfortunately, this photo also shows a common problem with them -- all clogged. These gutters, often nothing more than a curl in the eave with tar paper as a membrane, can be real problems. At some installations, the gutter is right above the wall of the home -- not out at the edge like with a normal gutter. When they get full, or leak for any other reason, that leads to water going down into the soffit, the wall or adjacent structure. The photo below is structural damage caused by rot around an overflowing integral gutter, same property shown above.
I would like to say that this is uncommon. Problem is, I almost always find damage, in varying degrees, when I find integral gutters. They might be fine in areas with light rain but they sure are not ideal in the Pacific NorthWET. A common repair, that makes sense, is to modify the roof and put in standard gutters that still require cleaning, but they keep leakage and water farther away from key parts of the home.
Steven L. Smith
Bellingham WA Home Inspections