Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving (a e2d z+b ed z+c)beta





http://functions.wolfram.com/01.03.21.0518.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d", " ", "z"]]]]], "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "+", "c"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "d"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["c", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "+", RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d", " ", "z"]]]]]]]]]], "-", RowBox[List["2", " ", SqrtBox["c"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "+", RowBox[List["2", " ", SqrtBox["c"], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "+", RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d", " ", "z"]]]]]]]]]]]], ")"]]]], SuperscriptBox["c", RowBox[List["3", "/", "2"]]]], "]"]]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]]]], SqrtBox["a"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["c", "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]]]], ")"]]]]]]]]]]], "]"]]]], SqrtBox["a"]]]], ")"]]]]]]]]










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> <msqrt> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> <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> <mi> d </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msqrt> <mi> c </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> d </mi> <mtext> </mtext> </mrow> <msup> <mi> c </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> c </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> b </mi> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2002-12-18





© 1998- Wolfram Research, Inc.