Integrate[Sin[a x^4] Sin[b x^2], {x, 0, Infinity}] == (-(1/4)) Pi Sqrt[b/(2 a)] Sin[b^2/(8 a) - (3 Pi)/8] BesselJ[1/4, b^2/(8 a)] /; Element[a, Reals] && Element[b, Reals]

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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> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> x </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 8 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </msub> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <pi /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sin /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> a </ci> <ci> &#8477; </ci> </apply> <apply> <in /> <ci> b </ci> <ci> &#8477; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18