Integrate[Cos[a z^2 + c], z] == (1/Sqrt[a]) (Sqrt[Pi/2] (Cos[c] FresnelC[Sqrt[a] Sqrt[2/Pi] z] - FresnelS[Sqrt[a] Sqrt[2/Pi] z] Sin[c]))

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", "c"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["a"]], RowBox[List["(", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "c", "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox["a"], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox["a"], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "c", "]"]]]]]], ")"]]]], ")"]]]]]]]]

<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> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> c </ci> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <sin /> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18