Integrate[DiracDelta[t - a] f[t], {t, -d, d}] == f[a] /; Inequality[-Infinity, LessEqual, -d, Less, a, Less, d, LessEqual, Infinity]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "d"]], "d"], RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List["t", "-", "a"]], "]"]], RowBox[List["f", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["f", "[", "a", "]"]]]], "/;", RowBox[List[RowBox[List["-", "\[Infinity]"]], "\[LessEqual]", RowBox[List["-", "d"]], "<", "a", "<", "d", "\[LessEqual]", "\[Infinity]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mi> d </mi> </msubsup> <mrow> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> &#8804; </mo> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> &lt; </mo> <mi> a </mi> <mo> &lt; </mo> <mi> d </mi> <mo> &#8804; </mo> <mi> &#8734; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </lowlimit> <uplimit> <ci> d </ci> </uplimit> <apply> <times /> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> f </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <ci> f </ci> <ci> a </ci> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <lt /> <ci> a </ci> <lt /> <ci> d </ci> <leq /> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "d_"]], "d_"], RowBox[List[RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List["t_", "-", "a_"]], "]"]], " ", RowBox[List["f", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["f", "[", "a", "]"]], "/;", RowBox[List[RowBox[List["-", "\[Infinity]"]], "\[LessEqual]", RowBox[List["-", "d"]], "<", "a", "<", "d", "\[LessEqual]", "\[Infinity]"]]]]]]]]

2001-10-29