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Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function > Involving exp > Involving ab zr+d z sin(c z)





http://functions.wolfram.com/01.06.21.0195.01









  


  










Input Form





Integrate[a^(b Sqrt[z] + d z) Sin[c z], z] == (-(1/2)) ((a^(b Sqrt[z] + d z) E^(I c z))/(c - I d Log[a]) + a^(b Sqrt[z] + d z)/(E^(I c z) (c + I d Log[a])) + b Sqrt[Pi] Log[a] ((I E^((b^2 Log[a]^2)/(4 ((-I) c - d Log[a]))) Erf[(-2 I c Sqrt[z] - b Log[a] - 2 d Sqrt[z] Log[a])/ (2 Sqrt[(-I) c - d Log[a]])])/(2 ((-I) c - d Log[a])^(3/2)) - (I E^((b^2 Log[a]^2)/(4 (I c - d Log[a]))) Erf[(2 I c Sqrt[z] - b Log[a] - 2 d Sqrt[z] Log[a])/ (2 Sqrt[I c - d Log[a]])])/(2 (I c - d Log[a])^(3/2))))










Standard Form





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MathML Form







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</apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["a_", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]], RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", "z"]]]]], RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], "+", RowBox[List["b", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Log", "[", "a", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "c", " ", SqrtBox["z"]]], "-", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", SqrtBox["z"]]], "-", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], ")"]]]]]], ")"]]]]]]]]










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





2002-12-18