Is there a way to integrate Azure Application Gateway logs to the Azure Log integration service. I can only see that Log integration seems to work for the VM logs, Security Center, Keyvault and Azure AD audit logs but nothing else.
How can we integrate other Azure cloud services(App gateway diagnostics etc.) with Azure Log Integration?
First, to answer the question about why you are don't see log integration for Application Gateway. Log Integration is only for resources like the VM, Security centre and other things you have mentioned. The Application Gateway is more like an appliance (like a tool that you use) rather than a service (that you consume). I think that is why they are separate and you are unable to mix them up.
I guess you have already seen this link about gateway diagnostics but I am putting it here anyway - https://learn.microsoft.com/en-us/azure/application-gateway/application-gateway-diagnostics
Perhaps, it is possible for you to take the logs from the Log Integration and then the logs from the gateway and combine them in a web app of your own design.