Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function > Involving exp > Involving ad z+e sinh(c zr+g)





http://functions.wolfram.com/01.19.21.0243.01









  


  










Input Form





Integrate[a^(d z + e) Sinh[c z^2 + g], z] == (1/(4 c)) ((a^e Sqrt[Pi] (Sqrt[-c] E^((d^2 Log[a]^2)/(2 c)) Erfi[(-2 c z + d Log[a])/(2 Sqrt[-c])] (Cosh[g] - Sinh[g]) + Sqrt[c] Erfi[(2 c z + d Log[a])/(2 Sqrt[c])] (Cosh[g] + Sinh[g])))/ E^((d^2 Log[a]^2)/(4 c)))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["d", " ", "z"]], "+", "e"]]], RowBox[List["Sinh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "c"]]], RowBox[List["(", RowBox[List[SuperscriptBox["a", "e"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "c"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["2", " ", "c"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "-", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]], "+", RowBox[List[SqrtBox["c"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "c", " ", "z"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "+", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<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> a </mi> <mrow> <mi> e </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> g </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> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mi> e </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <mi> c </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> g </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> g </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mi> c </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> c </mi> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> g </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> g </mi> <mo> ) </mo> </mrow> </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> a </ci> <apply> <plus /> <ci> e </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <sinh /> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> g </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <ci> e </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <cosh /> <ci> g </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sinh /> <ci> g </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <cosh /> <ci> g </ci> </apply> <apply> <sinh /> <ci> g </ci> </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["d_", " ", "z_"]], "+", "e_"]]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["a", "e"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "c"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["2", " ", "c"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "-", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]], "+", RowBox[List[SqrtBox["c"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "c", " ", "z"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "+", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", "c"]]]]]]]










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





2002-12-18