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http://functions.wolfram.com/01.06.21.1076.01
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Integrate[Sin[c z]^2/Sqrt[a z + b], z] ==
(1/(2 c)) (Sqrt[c/a] (2 Sqrt[c/a] Sqrt[b + a z] -
Sqrt[Pi] Cos[(2 b c)/a] FresnelC[(2 Sqrt[c/a] Sqrt[b + a z])/Sqrt[Pi]] -
Sqrt[Pi] FresnelS[(2 Sqrt[c/a] Sqrt[b + a z])/Sqrt[Pi]] Sin[(2 b c)/a]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], SqrtBox[RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "c"]]], RowBox[List["(", RowBox[List[SqrtBox[FractionBox["c", "a"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[FractionBox["c", "a"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["2", " ", SqrtBox[FractionBox["c", "a"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["2", " ", SqrtBox[FractionBox["c", "a"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], SqrtBox["\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"], "]"]]]]]], ")"]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mfrac> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </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 /> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"], SqrtBox[RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[FractionBox["c", "a"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[FractionBox["c", "a"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["2", " ", SqrtBox[FractionBox["c", "a"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["2", " ", SqrtBox[FractionBox["c", "a"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], SqrtBox["\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"], "]"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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