Integrate[z^2 E^(b z) Sin[c z], z] == (1/(b^2 + c^2)^3) (E^(b z) ((-c) (-4 b^3 z - 4 b c^2 z + b^4 z^2 + c^2 (-2 + c^2 z^2) + 2 b^2 (3 + c^2 z^2)) Cos[c z] + (-2 b^4 z + 2 c^4 z + b^5 z^2 + b c^2 (-6 + c^2 z^2) + 2 b^3 (1 + c^2 z^2)) Sin[c z]))

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "2"], SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], "3"]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["b", "3"], " ", "z"]], "-", RowBox[List["4", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["b", "4"], " ", "z"]], "+", RowBox[List["2", " ", SuperscriptBox["c", "4"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "5"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 4 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["b", "3"], " ", "z"]], "-", RowBox[List["4", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["b", "4"], " ", "z"]], "+", RowBox[List["2", " ", SuperscriptBox["c", "4"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "5"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], "3"]]]]]]

2002-12-18