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 power function > Involving power > Involving zalpha-1and arguments a zr+b





http://functions.wolfram.com/01.19.21.0129.01









  


  










Input Form





Integrate[z^4 Sinh[a Sqrt[z] + b], z] == (1/a^10) (2 (a Sqrt[z] (362880 + 60480 a^2 z + 3024 a^4 z^2 + 72 a^6 z^3 + a^8 z^4) Cosh[b + a Sqrt[z]] - 9 (40320 + 20160 a^2 z + 1680 a^4 z^2 + 56 a^6 z^3 + a^8 z^4) Sinh[b + a Sqrt[z]]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", "b"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["a", "10"]], RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["362880", "+", RowBox[List["60480", " ", SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List["3024", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["72", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["z", "3"]]], "+", RowBox[List[SuperscriptBox["a", "8"], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List["9", " ", RowBox[List["(", RowBox[List["40320", "+", RowBox[List["20160", " ", SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List["1680", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["56", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["z", "3"]]], "+", RowBox[List[SuperscriptBox["a", "8"], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", SqrtBox["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> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </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> <msup> <mi> a </mi> <mn> 10 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 72 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3024 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 60480 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 362880 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1680 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 20160 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 40320 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </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> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <sinh /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 10 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 72 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3024 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 60480 </cn> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 362880 </cn> </apply> <apply> <cosh /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1680 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 20160 </cn> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 40320 </cn> </apply> <apply> <sinh /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </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["z_", "4"], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["a_", " ", SqrtBox["z_"]]], "+", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["362880", "+", RowBox[List["60480", " ", SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List["3024", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["72", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["z", "3"]]], "+", RowBox[List[SuperscriptBox["a", "8"], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List["9", " ", RowBox[List["(", RowBox[List["40320", "+", RowBox[List["20160", " ", SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List["1680", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["56", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["z", "3"]]], "+", RowBox[List[SuperscriptBox["a", "8"], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]], SuperscriptBox["a", "10"]]]]]]










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





2002-12-18