Thursday, September 3, 2015

integration - Fresnel Integral?




Is cos(π2x2)dx a known integral ?




I found on the net something called Fresnel integral, but we didn't learn it, and it also somehow related to Euler, and we didn't touch the Euler stuff, so maybe I made a mistake before it while calculating the double integral:



Dsin(πx2y)dxdy

where D={(x,y)R2:xy3xy22}




So I wrote the D as a simple to x:



{22y1y3xy



and then I did the integral by x and got that troublesome integral.




So can I solve this problem without Fresnel integral or maybe i have some mistake on the way?




Answer



Yes, you can do without Fresnel integrals I give you the first step:
yy3sin(πx2y)dx=2πycos(πy22).


Then integrate with respect to y (if you do not see how to do that I give you the hint to change variable u=y2).


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