Wednesday, 26 November 2014

calculus - Evaluating the integral inti0nftyfracxsinrxa2+x2dx using only real analysis




Calculate the integral0xsinrxa2+x2dx=12xsinrxa2+x2dx,a,rR.
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=πeara.

Differentiating the both sides of equation above with respect to r yields
0ddr(cosrxx2+a2) dx=ddr(πeara)0xsinrxx2+a2 dx=(a)πeara0xsinrxx2+a2 dx=πear.
Done! :)


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