Editing Improper integral (section)

===Functions with both positive and negative values===
These definitions apply for functions that are non-negative.  A more general function ''f'' can be decomposed as a difference of its positive part <math>f_+=\max\{f,0\}</math> and negative part <math>f_-=\max\{-f,0\}</math>, so
:<math>f=f_+-f_-</math>
with <math>f_+</math> and <math>f_-</math> both non-negative functions.  The function ''f'' has an improper Riemann integral if each of <math>f_+</math> and <math>f_-</math> has one, in which case the value of that improper integral is defined by
:<math>\int_Af = \int_Af_+ - \int_A f_-.</math>
In order to exist in this sense, the improper integral necessarily converges absolutely, since
:<math>\int_A|f| = \int_Af_+ + \int_Af_-.</math><ref>{{harvnb|Cooper|2005|loc=p. 538}}: "We need to make this stronger definition of convergence in terms of |''f''(''x'')| because cancellation in the integrals can occur in so many different ways in higher dimensions."</ref><ref>{{harvnb|Ghorpade|Limaye|2010|loc=p. 448}}: "The relevant notion here is that of unconditional convergence." ... "In fact, for improper integrals of such functions, unconditional convergence turns out to be equivalent to absolute convergence."</ref>