Should I include my second passport in my UK Visa application?

If the application form asks for your travel history, it means you History of international travel, not where the passport with which the application was filed has been. Leaving things outside counts as lying on a visa application – what spells doom not only for that application, but also for their interactions with the UK visa and the immigration authorities for a long, long time afterwards.

However, you can not expect to present both passports with the visa application; That would nullify the purpose of having an additional passport. But it's probably a good idea to explain in your application. that you have a second passport, to avoid confusion that arises from the fact that the travel history you are describing should have resulted in more stamps than you can see in the passport you are presenting.