How to prove this equality with closed form?An exercise I'm trying to solve gives usF(y)&nbsp;=∫0∞sin⁡(xy)x(1+x2)&nbsp;dx&nbsp;and first asks us to prove that F(y) satisfies the ODE&nbsp;F&nbsp;″(y)−F(y)+π2=0, then find the closed form of the ODE and lastly prove that:∫0∞cos⁡(yx)1+x2&nbsp;dx&nbsp;=πe−y2using the closed form of the ODE.I've gone through the first two and for the closed form I got&nbsp;F(y)=c1e−y+c2ey+π2.But I'm having trouble with solving the last part of the exercise. With complex analysis, I can solve it, but I don't know what to do with the closed form. I'm sharing my original thoughts, but I get stuck:First, I noticed that the last integral is the derivative of F(y):F'(y)=∫0∞ddy(sin⁡(xy)x(1+x2))dx=∫0∞cos⁡(yx)1+x2dxAlso:&nbsp;F′(y)=−c1e−y+c2ey. Then:∫0∞cos⁡(yx)1+x2&nbsp;dx&nbsp;=−c1e−y+c2eyThen I'd need to show&nbsp;−c1e−y+c2ey=πe−y2&nbsp;and I get stuck here.Am I doing something wrong? Is this even solvable this way? Should I approach it differently?

Question

How to prove this equality with closed form?An exercise I&#039;m trying to solve gives usF(y)&amp;nbsp;=∫0∞sin⁡(xy)x(1+x2)&amp;nbsp;dx&amp;nbsp;and first asks us to prove that F(y) satisfies the ODE&amp;nbsp;F&amp;nbsp;″(y)−F(y)+π2=0, then find the closed form of the ODE and lastly prove that:∫0∞cos⁡(yx)1+x2&amp;nbsp;dx&amp;nbsp;=πe−y2using the closed form of the ODE.I&#039;ve gone through the first two and for the closed form I got&amp;nbsp;F(y)=c1e−y+c2ey+π2.But I&#039;m having trouble with solving the last part of the exercise. With complex analysis, I can solve it, but I don&#039;t know what to do with the closed form. I&#039;m sharing my original thoughts, but I get stuck:First, I noticed that the last integral is the derivative of F(y):F&#039;(y)=∫0∞ddy(sin⁡(xy)x(1+x2))dx=∫0∞cos⁡(yx)1+x2dxAlso:&amp;nbsp;F′(y)=−c1e−y+c2ey. Then:∫0∞cos⁡(yx)1+x2&amp;nbsp;dx&amp;nbsp;=−c1e−y+c2eyThen I&#039;d need to show&amp;nbsp;−c1e−y+c2ey=πe−y2&amp;nbsp;and I get stuck here.Am I doing something wrong? Is this even solvable this way? Should I approach it differently?

betazpvaf4 · Accepted Answer

Step 1If you proven the first parts already, then all you need is a limiting condition.F′(0)=∫0∞cos⁡(0)(1+x2)dx=π2and consider the change of variables z=xyStep 2F(y)=∫0∞sin⁡zz(1+z2y2)dz⇒limy→∞F(y)=∫0∞sin⁡zz:dz=π2The exponentially increasing term must be 0 since the limit exists at all, and then we haveF′(y)=πe−y2

How to prove this equality with closed form? An

Answered question

Answer & Explanation

New Questions in Calculus 2