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 FractionalPart

 http://functions.wolfram.com/04.05.21.0005.02

 Input Form

 Integrate[FractionalPart[t], {t, 0, a}] == (1/2) (FractionalPart[a]^2 - Sign[a] FractionalPart[a] + Abs[a])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a"], RowBox[List[RowBox[List["FractionalPart", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["FractionalPart", "[", "a", "]"]], "2"], "-", RowBox[List[RowBox[List["Sign", "[", "a", "]"]], " ", RowBox[List["FractionalPart", "[", "a", "]"]]]], "+", RowBox[List["Abs", "[", "a", "]"]]]], ")"]]]]]]]]

 MathML Form

 0 a frac ( t ) t 1 2 ( frac ( a ) 2 - sgn ( a ) frac ( a ) + "\[LeftBracketingBar]" a "\[RightBracketingBar]" ) t 0 a FractionalPart t 1 2 FractionalPart a 2 -1 Sign a FractionalPart a a [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a_"], RowBox[List[RowBox[List["FractionalPart", "[", "t_", "]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["FractionalPart", "[", "a", "]"]], "2"], "-", RowBox[List[RowBox[List["Sign", "[", "a", "]"]], " ", RowBox[List["FractionalPart", "[", "a", "]"]]]], "+", RowBox[List["Abs", "[", "a", "]"]]]], ")"]]]]]]]]

 Contributed by

 David W. Cantrell (2006)

 Date Added to functions.wolfram.com (modification date)

 2001-10-29