Logging your application with App Insights instead of Windows Azure Diagnostics

We've recently started deploying applications to Microsoft Azure. Unfortunately, we don't use Cloud Apps for our web apps, and instead use MSDeploy on the command line to enable automated deployment. Consequently, we don't have any .csdef files in which to implement Windows Azure Diagnostics (WAD).

Fortunately, there's a suitable replacement for applications in Azure App Service (rather than Cloud Services): Application Insights. Not only does this provide a TraceListener for tracing your application, but also provides a whole suite of other useful diagnostics, making this something that you should really be using anyway.  Using the links here and here, I was able to get up to speed with App Insights within an hour and a half, and implement a working solution for our application.