Calculate the integral∫∞0xsinrxa2+x2dx=12∫∞−∞xsinrxa2+x2dx,a,r∈R.
Edit: I was able to solve the integral using complex analysis, and now I want to try and solve it using only real analysis techniques.
Answer
It looks like I'm too late but still I wanna join the party. :D
Consider
∫∞0cosrxx2+a2 dx=πe−ara.
Differentiating the both sides of equation above with respect to r yields
∫∞0ddr(cosrxx2+a2) dx=ddr(πe−ara)−∫∞0xsinrxx2+a2 dx=(−a)πe−ara∫∞0xsinrxx2+a2 dx=πe−ar.
Done! :)
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