Is ∫cos(π2⋅x2)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:x≤y≤3√x∧y≥√22}
So I wrote the D as a simple to x:
{√22≤y≤1y3≤x≤y
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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