Integrate[Cos[b Sqrt[z]] Cos[c Sqrt[z]], z] == (Cos[(-b - c) Sqrt[z]] - (b + c) Sqrt[z] Sin[(-b - c) Sqrt[z]])/(b + c)^2 + (Cos[(-b + c) Sqrt[z]] + (-b + c) Sqrt[z] Sin[(-b + c) Sqrt[z]])/(-b + c)^2

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<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> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <cos /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <cos /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18