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 and algebraic functions > Involving exp and algebraic functions > Involving (a z+b)beta dzsinh(c z+e)





http://functions.wolfram.com/01.19.21.0405.01









  


  








Input Form





Integrate[(E^(p z) Sinh[c z])/Sqrt[a z + b], z] == (Sqrt[Pi] (E^((2 b c)/a) (c + p) Sqrt[((c - p) (b + a z))/a] Erfc[Sqrt[((c - p) (b + a z))/a]] + (c - p) Sqrt[-(((c + p) (b + a z))/a)] Erfc[Sqrt[-(((c + p) (b + a z))/a)]]))/E^((b (c + p))/a)/ (2 (c^2 - p^2) Sqrt[b + a z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], "a"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"]], " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]], " ", RowBox[List["Erfc", "[", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "-", SuperscriptBox["p", "2"]]], ")"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]], ")"]]]]]]]]








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> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> p </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> </msqrt> <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 /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <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> <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> <apply> <plus /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> </apply> </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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> <apply> <power /> <ci> a </ci> <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 /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <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> <plus /> <ci> c </ci> <ci> p </ci> </apply> <apply> <ci> Erfc </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <ci> Erfc </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], "a"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"]], " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]], " ", RowBox[List["Erfc", "[", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "-", SuperscriptBox["p", "2"]]], ")"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]]]]]]]]]


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





2002-12-18