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So I'm studying a course on measure theory and we've learnt that the Lebesgue integral of a real function is (loosely) defined as the total area over the x-axis minus the total area under the x-axis. This seems to me to be limited because these areas can both be infinite but their difference may be finite (such as sin(x)/x integrated from 0 to +inf) and, for me, this is a major failing for a definition of the integral. I was wondering whether there were any extensions to the Lebesgue integral that can handle these types of functions?