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Exp




Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function > Involving power > Involving zalpha-1and arguments a z





http://functions.wolfram.com/01.03.21.0067.01









  


  








Input Form





Integrate[z^5 E^(a z), z] == (E^(a z) (-120 + 120 a z - 60 a^2 z^2 + 20 a^3 z^3 - 5 a^4 z^4 + a^5 z^5))/ a^6








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "5"], SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "120"]], "+", RowBox[List["120", " ", "a", " ", "z"]], "-", RowBox[List["60", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["20", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["5", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "+", RowBox[List[SuperscriptBox["a", "5"], " ", SuperscriptBox["z", "5"]]]]], ")"]]]], SuperscriptBox["a", "6"]]]]]]








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> 5 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 5 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 120 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 120 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mfrac> </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'> 5 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 60 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 120 </cn> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> -120 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </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_", "5"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a_", " ", "z_"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "120"]], "+", RowBox[List["120", " ", "a", " ", "z"]], "-", RowBox[List["60", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["20", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["5", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "+", RowBox[List[SuperscriptBox["a", "5"], " ", SuperscriptBox["z", "5"]]]]], ")"]]]], SuperscriptBox["a", "6"]]]]]]


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





2002-12-18